Combinatorial Geometry with Applications to Field Theory, by L. Mao

8/14/2019 Combinatorial Geometry with Applications to Field Theory, by L. Mao

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LINFAN MAO

COMBINATORIAL GEOMETRY

WITH APPLICATIONS TO FIELD THEORY

A

B C

M K

K

LN-

+

InfoQuest

2009



Linfan MAO

Academy of Mathematics and SystemsChinese Academy of SciencesBeijing 100080, P.R.China

Email: [email protected]

Combinatorial Geometry

with Applications to Field Th



This book can be ordered in a paper bound reprint from:

Books on DemandProQuest Information & Learning

(University of Microlm International)300 N.Zeeb RoadP.O.Box 1346, Ann ArborMI 48106-1346, USATel:1-800-521-0600(Customer Service)

http://wwwlib.umi.com/bod

Peer Reviewers:

F.Tian, Academy of Mathematics and Systems, Chinese Academy o

jing 100080, P.R.China.J.Y.Yan, Graduate Student College, Chinese Academy of Sciences, P.R.China.R.X.Hao, Department of Applied Mathematics, Beijing Jiaotong Un100044, P.R.China.W.L.He, Department of Applied Mathematics, Beijing Jiaotong Uni100044, P.R.China.

Copyright 2009 by InfoQuest and Linfan Mao



Preface

Anyone maybe once heard the proverb of the six blind mewhich these blind men were asked to determine what an elephdifferent parts of the elephant’s body. The man touched its legor tusk claims that the elephant is like a pillar, a rope, a tree wall or a solid pipe, respectively. Each of them insisted his viinto an endless argument. All of you are right ! A wise manare you telling it differently is because each one of you touchthe elephant. So, actually the elephant has all those features

After read this meaningful proverb, we should ask ourse

What is its implication in philosophy ?

What is its meaning for understanding of the WORLD One interesting implication of this proverb is that an

a union of those claims of the six blind men, i.e., a Smaranderlying a combinatorial structures . The situation for one rea



ii Linfan Mao: Combinatorial Geometry with Applications t

mathematics and physics by combinatorics, i.e., mathematical comning in 2004 when I was a post-doctor of Chinese Academy of System Science. Now, it nally bring about this self-contained booGeometry with Applications to Field Theory , includes combinatoalgebraic combinatorics, differential Smarandache manifolds, combitial geometry, quantum elds with dynamics, combinatorial elds wand so on.

Contents in this book are outlined following.

Chapters 1 and 2 are the fundamental of this book. In Chapintroduce combinatorics with graphs, such as those of Boolean algpartially ordered or countable sets, graphs and combinatorial enumare useful in following chapters.

Chapter 2 is the fundamental of mathematical combinatorics, tion of combinatorial notion to mathematical systems, i.e., combinparticularly algebraic systems. These groups, rings and modules we

a combinatorial one. We also consider actions of multi-groups on extends a few well-known results in classical permutation groups.

Chapter 3 is a survey of Smarandache geometries. For intrtial Smarandache manifolds, we rst present topological spaces wgroups, covering space and simplicial homology group, Euclidean

tial forms in Rn

and the Stokes theorem on simplicial complexes.Smarandache geometries, map geometries and pseudo-Euclidean sducing differential structure on Smarandache manifolds, we discumanifold, principal ber bundles and geometrical inclusions in diff



Preface

chapter.

Chapters 5 and 6 form the main parts of combinatoritry, which provides the fundamental for applying it to physicChapter 5 discuss tangent and cotangent vector space, tensor ferentiation on combinatorial manifolds, connections and curcombinatorial Riemannian manifolds, integrations and the genand Gauss’ theorem, and so on. Chapter 6 contains three patrates on combinatorial submanifold of smooth combinatoriadamental equations. The second generalizes topological grouexample Lie multi-groups. The third is a combinatorial gene

ber bundled to combinatorial manifolds by voltage assignmprovides the mathematical fundamental for discussing combinChapter 8.

Chapters 7 and 8 introduce the applications of combinatorFor this objective, variational principle, Lagrange equations

equations in mechanical elds, Einstein’s general relativity wMaxwell eld and Abelian or Yang-Mills gauge elds are inApplying combinatorial geometry discussed in Chapters 4elds to combinatorial elds under the projective principlea combinatorial eld is invariant under a projection on its

Then, we show how to determine equations of combinatordensity, to solve equations of combinatorial gravitational eldscombinatorial gauge basis and elds, · · ·.

This book was began to write in October, 2006. Many co



iv Linfan Mao: Combinatorial Geometry with Applications t

Junliang Cai, Rongxia Hao, Wenguang Zai, Goudong Liu, WeiWei for their kindly helps and often discussing problems in mathemPartially research results of mine were reported at Chinese Academy& System Sciences, Beijing Jiaotong University, East-China NormaHunan Normal University in past years. Some of them were also repand 3rd Conference on Combinatorics and Graph Theory of China The 3rd and 4th International Conference on Number Theory andProblems of Northwest of China in 2007 and 2008. My sincerely thto these audiences discussing mathematical topics with me in thes

Of course, I am responsible for the correctness all of these m

here. Any suggestions for improving this book and solutions for this book are welcome.

L.FAMJul



Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 1. Combinatorics with Graphs . . . . . . . . . . . . . .

§1.1 Sets with operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.1 Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.2 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.3 Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.4 Multi-set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§1.2 Partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 .2 .1 Par t ia l ly ordered se t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.2 Multi-poset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§1.3 Countable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 . 3 . 1 M a p p i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 2 Countable set



vi Linfan Mao: Combinatorial Geometry with Applications t

1.5.1 Enumeration principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.5.2 Inclusion-exclusion principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 .5 .3 Enumerat ing mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.5.4 Enumerating labeled graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§1.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2. Fundamental of Mathematical Combinatorics

§2.1 Combinatorial systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1.1 Proposition in lgic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1.2 Mathematical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1.3 Combinatorial system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§2.2 Algebraic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 .2 .1 Algebra ic sys tem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.2 Associative and commutative law . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.3 Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.4 Isomorphism of Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.5 Homomorphism theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§2.3 Multi-operation systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3.1 Multi-operation systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.2 Isomorphism of mult i-systems. . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.3 Distribute law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.4 Multi-group and multi-ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



Contents

§2.6 Combinatorial algebraic systems . . . . . . . . . . . . . . . . . . . . . .

2 .6 .1 Algebraic mul t i -sys tem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 .6 .2 Diagram of mul t i -sys tem. . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6.3 Cayley diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§2.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3. Smarandache manifolds . . . . . . . . . . . . . . . . . . .

§3.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.1 Topological space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.2 Metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.3 Fundamental group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.4 Covering space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.5 Simplicial homology group . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 .1 .6 Topological manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§3.2 Euclidean geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 .2 .1 Euc l idean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.2 Linear mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.3 Differential calculus on R n . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.4 Differential form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2.5 Stokes’ theorem on simplicial complex . . . . . . . . . . . . . . .

§3.3 Smarandache manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



viii Linfan Mao: Combinatorial Geometry with Applications t

§3.5 Pseudo-manifold geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§3.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4. Combinatorial Manifolds . . . . . . . . . . . . . . . . . . . . . . .

§4.1 Combinatorial space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.1.1 Combinatorial Euclidean space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.1.2 Combinatorial fan-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.1.3 Decomposition space into combinatorial one. . . . . . . . . . . . . . . .

§4.2 Combinatorial manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 .2 .1 Combinator ia l manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2.2 Combinatorial submanifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2.3 Combinatorial equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2.4 Homotopy class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2.5 Euler-Poincare characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§4.3 Fundamental groups of combinatorial manifolds . . . . . . . . .. . . .

4 . 3 . 1 Re t r ac t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3.2 Fundamental d-group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3.3 Homotopy equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§4.4 Homology groups of combinatorial manifolds . . . . . . . . . . . . . .4.4.1 Singular homology group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.4.2 Relative homology group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 4 3 Exact chain



x Linfan Mao: Combinatorial Geometry with Applications t

5.7.2 Combinatorial Gauss’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§5.8 Combinatorial Finsler geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.8.1 Combinatorial Minkowskian norm. . . . . . . . . . . . . . . . . . . . . . . . . . .

5.8.2 Combinatorial Finsler geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 .8 .3 Geometr ica l inclus ion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§5.9 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 6. Combinatorial Riemannian Submanifoldswith Principal Fiber Bundles . . . . . . . . . . . . . . . . . .

§6.1 Combinatorial Riemannian submanifolds . . . . . . . . . . . . . . . . . . . .

6.1.1 Fundamental formulae of submanifold. . . . . . . . . . . . . . . . . . . . . . .

6.1.2 Local form of fundamental formulae. . . . . . . . . . . . . . . . . . . . . . . .

§6.2 Fundamental equations on combinatorial submanifolds . . . . . .

6.2.1 Gauss equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.2.2 Codazzi equaton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2.3 Ricci equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2.4 Local form of fundamental equation. . . . . . . . . . . . . . . . . . . . . . . .

§6.3 Embedded combinatorial submanifolds . . . . . . . . . . . . . . . . . . . . . .

6.3.1 Embedded combinatorial submanifold. . . . . . . . . . . . . . . . . . . . . . .

6.3.2 Embedded in combinatorial Euclidean space . . . . . . . . . . . . . . .

§6.4 Topological multi-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



Contents

6.5.1 Principal ber bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.5.2 Combinatorial Principal ber bundle . . . . . . . . . . . . . . . . . .6.5.3 Automorphism of principal ber bundle . . . . . . . . . . . . . . .6 .5 .4 Gauge t ransformat ion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.5.5 Connection on principal ber bundle . . . . . . . . . . . . . . . . . .6.5.6 Curvature form on principal ber bundle . . . . . . . . . . . . . .

§6.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 7. Fields with Dynamics . . . . . . . . . . . . . . . . . . . . . .

§7.1 Mechanical Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1.1 Particle Dynamic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 .1 .2 Var ia t ional Pr inciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 .1 .3 Hamil tonian pr inciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.1.4 Lagrange Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.1.5 Hamiltonian Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.1.6 Conservation Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.1.7 Euler-Lagrange Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§7.2 Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2.1 Newtonian Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . . .

7.2.2 Einstein s Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.2.3 Einstein Gravitational Field. . . . . . . . . . . . . . . . . . . . . . . . . . . .7.2.4 Limitation of Einstein s Eq u a t i on . . . . . . . . . . . . . . . . . . .7 .2 .5 Schwarzschi ld Metr ic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



xii Linfan Mao: Combinatorial Geometry with Applications t

§7.4 Gauge Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.4.1 Gauge Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.4.2 Maxwell Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.4.3 Weyl Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.4.4 Dirac Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.4.5 Yang-Mills Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.4.6 Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.4.7 Geometry of Gauge Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§7.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 8. Combinatorial Fields with Applications . . . . .

§8.1 Combinatorial Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 .1 .1 Combinator ia l Fie ld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.1.2 Combinatorial Conguration Space . . . . . . . . . . . . . . . . . . . . . . . . .

8.1.3 Geometry on Combinatorial Field. . . . . . . . . . . . . . . . . . . . . . . . . . .8.1.4 Covariance Principle in Combinatorial Field. . . . . . . . . . . . . . . .

§8.2 Equation of Combinatorial Field . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.2.1 Lagrangian on Combinatorial Field . . . . . . . . . . . . . . . . . . . . . . . . .8.2.2 Hamiltonian on Combinatorial Field . . . . . . . . . . . . . . . . . . . . . . . .8.2.3 Equation of Combinatorial Field . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.2.4 Tensor Equation on Combinatorial Field . . . . . . . . . . . . . . . . . . . .

§8.3 Combinatorial Gravitational Fields. . . . . . . . . . . . . . . . . . . . . . . . . .



Contents

8.4.3 Combinatorial Gauge Field . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.4.4 Geometry on Combinatorial Gauge Field. . . . . . . . . . . . . .8.4.5 Higgs Mechanism on Combinatorial Gauge Field . . . . .

§8.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5.1 Many-Body Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 . 5 . 2 Cos m ol ogy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.5.3 Physical Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.5.4 Economical Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



xiv Linfan Mao: Combinatorial Geometry with Applications t

No object is mysterious. The mystery is our

By Elizabeth, a Brit



CHAPTER 1.

Combinatorics with Graphs

For catering the need of computer science, combinatorics havewith many important results produced in the past century. Tthe essence of combinatorics ? In fact, it is in a combinatori

namely, combining different elds into a unifying one withoutis why only abstract notations are considered in combinatoricster, we introduce main ideas and techniques in combinatorics mmathematical combinatorics in the follow-up chapters. Certaalso viewed as a brief introduction to combinatorics and grap



2 Chap.1 Com

§1.1 SETS WITH OPERATIONS

1.1.1 Set. A set S is a collection of objects with properties P i ,by

S = {x|x posses properties P i , 1 ≤ i ≤s}.

For examples,

A = {natural numbers diviable by a prime p},

B = {cities with persons more than 10 million in the w

are two sets by denition. In philosophy, a SET is a category coThat is why we use conceptions of SET or PROPERTY withoudistinguish them just by context in mathematics sometimes.

An element x possessing properties P i , 1 ≤ i ≤ s is said an S , denoted by x∈S . Conversely, an element y without all propeis not an element of S , denoted y∈S . Denoted by |S | the cardIn the case of nite set, |S | is the number of elements in S .

Let S 1 and S 2 be two sets. If for∀x∈S 1, there must be x

that S 1 is a subset of S 2 or S 1 is included in S 2, denoted by SS 1 of S 2 is proper , denoted by S 1 ⊂S 2 if there exists an elemy∈S 1 hold. Further, the void (empty) set ∅, i.e., |∅|= 0 is a sudenition.



Sec.1.1 Sets with Operations

Notice that the relation of inclusion ⊆is reexive, asymmetric. Otherwise, by Theorem 1 .1, if S 1 ⊆S 2 and S

nd that S 1 = S 2. In summary, the inclusion relation ⊆following properties:

Reexive: For any S , S

⊆

S ;

Antisymmetric : If S 1⊆S 2 and S 2⊆S 1, then S 1 = S

Transitive : If S 1⊆S 2 and S 2⊆S 3, then S 1 = S

A set of cardinality i is called an i-set . All subsets of aset P (S ), called the power set of S . For a nite set S , we subsets.

Theorem 1.1.2 Let S be a nite set. Then

|P (S )| = 2 |S | .

Proof Notice that for any integer i, 1 ≤ i ≤ |S |, th

isomorphic subsets of cardinality i in S . Therefore, we nd

|P (S )

|=

|S |

i=1

|S |i

= 2 |S | .

1.1.2 Operations. For subsets S, T in a power set P (S

them can be introduced as follows.



4 Chap.1 Com

Idempotent : X X = X and X X = X ;

Commutative : X T = T X and X T = T X ;

Associative : X (T R) = ( X T ) R and X (T R

Distributive : X (T R) = ( X T ) (X R) and

X (T R) = ( X T ) (X R).

These idempotent, commutative and associative laws can be ately by denition. For the distributive law, let x∈X (T R

R). Then x∈X or x∈T R, i.e., x∈T and x∈R. Now if xx

∈

X

∪

T and x

∈

X

∪

R. Whence, we get that x

∈

(X T ) (Xx∈T R, i.e., x∈T and x∈R. We also get that x∈(X T )

Conversely, for ∀x ∈(X T ) (X R), we know that xX R, i.e., x∈X or x∈T and x∈R. If x∈X , we get thatIf x∈T and x∈R, we also get that x∈X (T R). Therefor(X T ) (X R) by denition.

Similar discussion can also veries the law X (T R) = ( X

Theorem 1.1.3 Let S be a set and X, T ∈P (S ). Then condit

equivalent.

(i) X ⊆T ;

(ii ) X ∩T = X ;

(iii ) X ∪T = T .

Proof The conditions (1) ⇒(2) and (1) ⇒(3) are obvious. N




X = {y | y∈S but y∈X }.

Then we know three laws on complementation of a set folloand intersection.

Complementarity : X ∩X = ∅and X ∪X = S ;

Involution : X = X ;

Dualization : X ∪T = X ∩T and X ∩T = X ∪These complementarity and involution laws can be imm

inition. For the dualization, let x ∈X ∪T . Then x ∈S

x ∈X and x ∈T . Whence, x ∈X and x ∈T . Therefore

∀x∈X ∩T , there must be x∈X and x∈T , i.e., x∈S

Hence, x∈X ∪T . This fact implies that x∈X ∪T . By X ∪T = X ∩T . Similarly, we can also get the law X ∩T =

For two sets S and T , the Cartesian product S

×T of S

all ordered pairs of elements ( a, b) for∀a∈S and ∀b∈T , i.

S ×T = {(a, b)|a∈S, b∈T }.

A binary operation ◦ on a set S is an injection mapping ◦ :

a subset R of S ×S is called a binary relation on S , and foraRb that a has relation R with b in S . A relation R on S is

Reexive: aRa for∀a∈S ;

S t i Rb i li bR f ∀ b∈S



6 Chap.1 Com

a∨b = b∨a, a∧b = b∧a,

and the associative laws

a∨(b∨c) = ( a∨b)∨c, a∧(b∧c) = ( a∧b)∧(ii ) The absorption laws

a∨(a∧b) = a∧(a∨b) = a.

(iii ) The distributive laws, i.e.,

a∨(b∧c) = ( a∧b)∨(a∧c), a∧(b∨c) = ( a∧b)∨

(iv) There exist two universal bound elements O, I in B su

O

∨

a = a, O

∧

a = O, I

∨

a = I, I

∧

a = a.

(v) There is a 1 −1 mapping ς : a →a obeyed laws

a∨a = I, a∧a = O.

Now choose operations

∪

=

∨

,

∩=

∧

and universal boundsP (S ). We know that

Theorem 1.1.4 Let S be a set. Then the power set P (S ) formsunder these union, intersection and complement operations.




Law B2 For ∀a, b∈B , a∨b = b if and only if a∧b = a.

In fact, if a∨b = b, then a∧b = a∧(a∨b) = a by tConversely, if a∧b = a, then a∨b = ( a∧b)∨b = b byabsorption laws.

Law B3 These equations a∨x = a∨y and a∧x = a∧y togCertainly, by the absorption, distributive and commutativ

x = x∧(a∨x) = x∧(a∨y)

= ( x∧a)∨(x∨y) = ( y∧x)∨(y∨= y∧(x∨a) = y∧(y∨a) = y.

Law B4 For ∀x, y∈B ,

x = x, (x

∧

y) = x

∨

y and (x

∨

y) =

Notice that x∧x = x∧x = O and x∨x = x∨xcomplement a is unique for ∀a ∈

B . We know that x = xassociative laws, we nd that

(x∧y)∧(x∨y) = ( x∧y∧x)∨(x∧y∧y= (( x∧x)∧y)∨(x∧(y

= ( O∧y)∨(x∧O) = O



8 Chap.1 Com

For variables x1, x2, · · ·, xn in B , polynomials f (x1, x2, · · ·,operations ∨and ∧are called Boolean polynomials . Each Boolea canonical form ensured in the next result.

Theorem 1.1.5 Any Boolean polynomial in x1, x2, · · ·, xn can bO or to join of some canonical forms

p1∧P 2∧ · · ·∧ pn ,

where each pi = xi or xi .

Proof According to the denition of Boolean algebra and laws

ical form for a Boolean polynomial, for example, f (x1, x2, x3) = x(x2∨x1), can be gotten by programming following.

STEP 1. If any complement occurs outside any parenthesis inmoved it inside by Law B4.

After all these complements have been moved all the way insidinvolving only vees and wedges action on complement and uncomThus, in our example: f (x1, x2, x3) = [x1∧x3∧(x2∨x3)]∨(x2∧STEP 2. If any ∧stands outside a parenthesis which contains a be moved inside by applying the distributive law.

There result a polynomial in which all meets

∧

are formed i.e., a join of terms in which each term is a meet of complement aletters. In the above example, f (x1, x2, x3) = ( x1∧x3∧x2)∨(x1∧STEP 3. If a letter y appears twice in one term, omit one occurrenc




Thus in our example, we nally get its canonical form fx3)∨(x1∧x2∧x3)∨(x1∧x2∧x3).

This completes the proof.

Corollary 1.1.1 There are 2n canonical forms and 2 2nB

variable x1, x2,

· · ·, xn in a Boolean algebra B with

|B

| ≥n

Dening a mapping η : B → {0, 1}by η(xi) = 1 or 0 a pi = x i in Theorem 1.1.5, we get a bijection between these Bvariable x1, x2, · · ·, xn and the set of all 2n n-digit binary numin the proof of Theorem 1.5, we have

η(f (x1, x2, x3)) = 010 , 111, 110.

1.1.4 Multi-Set. Considering the importance of Smaranmodern sciences, we discuss multi-sets as a preparing step in

For an integer n ≥1, a multi-set X is a union of sets X

two by two. Examples of multi-sets can be found in the follo

L = R T,

where R = {integers}, T = {polyhedrons }.

G = G1 G2 G3,

where G1 = {grvaitional eld }, G2 = {electric eld}and G3

denition, a multi-set is also a set only with a union structuri l h i h



10 Chap.1 Com

{sets with cardinality ≥2}= {multi −sets}.

This equality can be characterized more accurately by introducing parameters.

Theorem 1.1.7 For a set R with cardinality ≥ 2 and integers kexist k sets R1, R2, · · ·, Rk distinct two by two such that

R =k

i=1

R i

with

|k

i=1

R i | = s

if and only if

| R | ≥k + s.

Proof Assume there are sets k sets R1, R2, · · ·, Rk distinct two R =

k

i=1R i and |

k

i=1R i | = s. Notice that for any sets X and Y w

|X Y

|=

|X

|+

|Y

|and there is a subset

k

(R i \ (k

R t \ R i )) (k

R i)⊆

k

R i




= |k

i=1

(R i \ (k

t=1

R t \ R i))|+ |≥ k + s.

Now if | R | ≥k + s, let

{a1, a2, · · ·, ak , b1, b2, · · ·, bs}⊆ R

with a i = a j , bi = b j if i = j . Construct sets

R1 = {a2, · · ·, ak , b1, b2, · · ·, bs},

R2 = R \ {a2},

R3 = R \ {a3},

· · · · · · · · · · · · · · · · · ·,

Rk = R \ {ak}.

Then we get that

R =k

i=1

R i

and



Sec.1.2 Partially Ordered Sets

Two distinct elements x any y in a poset (X, P ) are calx < y or y < x , and incomparable otherwise. A poset in ware comparable is called a chain or ordered set , and one in are comparable is called an antichain or unordered set .

A subposet of a poset (X, P ) is a poset (Y, Q) in whic

restriction of P to Y ×Y . Two posets ( X, P ) and (X , P if there is a one-to-one correspondence τ : X →X such tonly if τ (x) ≤τ (y) in P . A poset (Y, Q) is said to be embedby (Y, Q)⊆(X, P ) if (Y, Q) is isomorphic to a subposet of orders P and Q on a set X , we call Q an extension of P

extension of P if Q is a chain. It is obvious that any poextension and the intersection of all linear extension of P isbe restated as follows:

for any two incomparable elements x and y in a poset (Xextension of P in which x < y , and another in which y < x

Denote a linear order L : x1 ≤x2 ≤ · · · ≤xn by L : [x1,poset (X, P ), a realizer {L1, L2, · · ·, L t}of P is a collectiowhose intersection is P , i.e., x < y in P if and only if x < yThe it dimension dim( X, P ) of a poset (X, P ) is dened toof realters R of P and the rank rank( X, P ) of (X, P ) to be

realizers R in which there are no proper subset of R is agaFor example, dim( X, P ) = 1 or rank( X, P ) = 1 if and ondim( X, P ) = 2 if it is an n-element antichain for n ≥2. Foinnite family, called the standard n-dimensional poset S0

n w



14 Chap.1 Com

Notice that if i = j , then b j < a i < bi < a j in Li , and bi

L j for any integers i,j, 1 ≤ i, j ≤n. Whence, R is a realizer of SdimS0

n ≤n.

Now if R∗ is any realizer of S0n , then for each k = 1 , 2, · · ·, n,

elements of R∗must have ak < bk , and‘furthermore, we can easily

no linear extensions L of S0n such that a i < bi and a j < b j for two This fact enables us to get that dim S0

n ≥n.

Therefore, we have dim S0n = n.

For rank S0n = n, notice that rank S0

n ≥dimS0n ≥n. Now obs

R of linear extension of S0n is a realizer if and only if , for i = 1 , 2,

a L i∈R at least such that a i < bi . Hence, n is also an upper bou

1.2.2 Multi-Poset. A multi-poset (X, P ) is a union of posets

· · ·, (X s , P s ) distinct two by two for an integer s ≥2, i.e.,

(X, P ) =s

i=1

(X i , P i),

also call it an s-poset. If each (X i , P i) is a chain for any integers it an s-chain. For a nite poset, we know the next result.

Theorem 1.2.2 Any nite poset (X, P ) is a multi-chain.

Proof Applying the induction on the cardinality

|X

|. If

|X

|is obvious. Now assume the assertion is true for any integer |X |case of |X | = k + 1.

Choose a maximal element a1∈X . If there are no elements



Sec.1.2 Partially Ordered Sets

is a maximal sequence such that a i+1 is covered by a i in (X, P a2, a1] is a chain. Consider (X \ {a1, a2, · · ·, a t− 1, a t}, P ).

|X \ {a1, a2, · · ·, a t− 1, a t}| ≤k. By the induction assumptioWhence,

(X, P ) = ( X \ {a1, a2, · · ·, a t− 1, a t}, P )

is also a multi-chain. In conclusion, we get that ( X, P ) is a of |X | = k + 1. By the induction principle, we get that ( X,any X with |X | ≥1.

Now consider the inverse problem, i.e., when is a multi

conditions in the following result.

Theorem 1.2.3 An s-poset (X, P ) =s

i=1(X i , P i) is a pose

integer i,j, 1 ≤ i, j ≤s, (x, y)∈P i and (y, z )∈P j imply tha

Proof Let (X, P ) be a poset. For any integer i,j, 1 ≤ i

and (y, z )∈P j also imply (x, y), (y, z )∈P . By the transitknow that ( x, z )∈P .

On the other hand, for any integer i,j, 1 ≤ i, j ≤s, if (ximply that ( x, z )∈P , we prove (X, P ) is a poset. Certainly,these reexive laws, antisymmetric laws and transitive laws

is divided into three discussions.(i) For ∀x∈X , there must exist an integer i, 1 ≤ i ≤denition. Whence, ( x, x )∈P i . Hence, (x, x )∈P , i.e., the (X, P ).



16 Chap.1 Com

Certainly, we can also nd more properties for multi-posets conditions. For example, construct different posets by introducinorders in a multi-poset. All these are referred to these readers itopics.

§1.3 COUNTABLE SETS

1.3.1 Mapping. A mapping f from a set X to Y is a subset ofor∀x∈X , |f (∩({x} ×Y )| = 1, i.e., f ∩({x} ×Y ) only has one

we denote a mapping f from X to Y by f : X →Y and f (x) the sof the unique element of f ∩({x}×Y ), called the image of x undenote all mappings from X to Y by Y X .

Let f : X →Y be a mapping. For any subsets U ⊆X andimage f (U ) of U under f to be

f (U ) = {f (u)| for ∀u∈U }and the inverse f − 1(V ) of V under f to be

f − 1(V ) = {u∈X |f (u)∈V }.

Generally, for U ⊆X , we have

U ⊆f − 1(f (U ))



Sec.1.3 Countable Sets

where, xi , yi ∈X and xi = x j , yi = y j if i = j for 1 ≤ i, jX = {1, 2, 3, 4, 5, 6}. Then

1 2 3 4 5 6 7 82 3 5 6 1 4 8 7

is a permutation. All permutations of X form a set, denoted on X is a particular permutation 1X ∈ (X ) given by 1X (x

For three sets X, Y and Z , let f : X →Y and h : Y →a mapping h ◦ f : X →Z , called the composition of f and h

h

◦f (x) = h(f (x))

for∀x∈X . It can be veried immediately that

(h ◦ f )− 1 = f − 1 ◦h− 1

by denition. We have a characteristic for bijections from Xoperations.

Theorem 1.3.1 A mapping f : X →Y is a bijection if anmapping h : Y →X such that f ◦h = 1Y and h ◦ f = 1X .

Proof If f is a bijection, then for ∀y ∈Y , there is that f (x) = y. Dene a mapping h : Y →X by h(y) =correspondent x. Then it can be veried immediately that

f ◦h = 1Y and h ◦f = 1X .

Now if there exists a mapping h : Y →X such that f ◦l i h f i j i d i j i O h i if f



18 Chap.1 Com

Whence, h ◦ f = 1X . Contradicts the assumption again.This completes the proof.

1.3.2 Countable Set. For two sets X and Y , the equality X |Y have the same cardinality means that there is a bijection f froX is said to be countable if it is bijective with the set Z of natuknow properties of countable sets and innite sets following.

Theorem 1.3.2(Paradox of Galileo) Any countable set X has proper subset of itself, i.e., the cardinal of a set maybe equal to it

Proof Since X is countable, we can represent the set X by

X = {xi|1 ≤ i ≤+ ∞}.Now choose a proper subset X = X \ {x1}and dene a b

X \ {x1}by

f (xi) = xi+1

for any integer i, 1 ≤ i ≤+ ∞. Whence, |X \ {x1}|= |X |.Theorem 1.3.3 Any innite set X contains a countable subset.

Proof First, choose any element x1∈X . From X \{x1}, theelement x2 and from X \{x1, x2}a third element x3, and so on. S

for any integer n, X \ {x1, x2, · · ·, xn}can never be empty. Whenchoose an new element xn +1 in the set X \ {x1, x2, · · ·, xn}. Thnever stop until we have constructed a subset X = {xi|1 ≤ i ≤countable subset X of X .



Sec.1.4 Graphs

countable subset X = {x1, x2, ···}. Now dene a bijection fsubset X \ {x1}by

f (x) =xi+1 , if x = xi∈X ,

x, if x∈X \ X .

Whence, X has a bijection with a proper subset X \ {x1}o

§1.4 GRAPHS

1.4.1 Graph. A graph G is an ordered 3-tuple ( V, E ; I ), w

V = ∅and I : E →V ×V . Call V the vertex set and E theby V (G) and E (G), respectively. An elements v∈V (G) is ine ∈E (G) if I (e) = ( v, x) or (x, v) for an x ∈V (G). Usfor∀u, v∈V , G is called a graph, otherwise, a directed grau →v on each edge (u, v).

The cardinal numbers of |V (G)| and |E (G)| are calledgraph G, denoted by |G| and ε(G), respectively.

Let G be a graph. It be can represented by locating eapoint p(u), p(u) = p(v) if u = v and an edge (u, v) by a c p(u) and p(v) on a plane R 2, where p : G →P is a mapping

For example, a graph G = ( V, E ; I ) with V =

{v1, v2, v3,

e6, e7, e8, e9, e10}and I (ei) = ( vi , vi), 1 ≤ i ≤ 4; I (e5) = ( v1

(v3, v4) = ( v4, v3), I (e6) = I (e7) = ( v2, v3) = ( v3, v2), I (e8

(v1, v4) can be drawn on a plane as shown in Fig.1 .4.1



20 Chap.1 Com

Let G = ( V, E ; I ) be a graph. For ∀e∈E , if I (e) = ( u, u )called a loop. For non-loop edges e1, e2∈E , if I (e1) = I (e2), themultiple edges of G. A graph is simple if it is loopless without mI (e) = ( u, v) implies that u = v, and I (e1) = I (e2) if e1 = e2 forthe case of simple graphs, an edge ( u, v) is commonly abbreviated

A walk of a graph G is an alternating sequence of vertices and e· · ·, en , un 1 with ei = ( u i , u i+1 ) for 1 ≤ i ≤n. The number n is cthe walk . A walk is closed if u1 = un +1 , and opened , otherwise. sequence v1e1v1e5v2e6v3e3v3e7v2e2v2 is a walk in Fig.1.3.1. A waledges are distinct and a path if all the vertices are distinct also.called a circuit usually.

A graph G = ( V, E ; I ) is connected if there is a path connectinin this graph. In a graph, a maximal connected subgraph is callA graph G is k-connected if removing vertices less than k fromconnected graph. Let G be a graph. For ∀u ∈V (G), the neighbthe vertex u in G is dened by N G (u) =

{v

|∀(u, v)

∈

E (G)

}. The

|N G (u)| is called the valency of vertex u in G and denoted by ρG (uρG (v) = 0 is an isolated vertex and ρG (v) = 1 a pendent vertex all vertices valency of G as a sequence ρG (u) ≥ ρG (v) ≥ ·· · sequence the valency sequence of G. By enumerating edges in Eequality is obvious.

u∈V (G)

ρG (u) = 2 |E (G)|.

A graph G with a vertex set V(G) = {v1 v2 vp}and an



Sec.1.4 Graphs

Let G1 = ( V 1, E 1; I 1) and G2 = ( V 2, E 2; I 2) be two grapdenoted by G1 = G2 if V 1 = V 2, E 1 = E 2 and I 1 = I 2. mapping φ : E 1 →E 2 and φ : V 1 →V 2 such that φI 1(e) =the convention that φ(u, v) = ( φ(u), φ(v)), then we say thG2, denoted by G1 ∼= G2 and φ an isomorphism between G

graphs H 1, H 2, this denition can be simplied by ( u, v)(φ(u), φ(v))∈I 2(E 2) for∀u, v∈V 1.

For example, let G1 = ( V 1, E 1; I 1) and G2 = ( V 2, E 2; I 2)

V 1 = {v1, v2, v3},

E 1 =

{e1, e2, e3, e4

},

I 1(e1) = ( v1, v2), I 1(e2) = ( v2, v3), I 1(e3) = ( v3, v1), I

and

V 2 = {u1, u2, u3},

E 2 =

{f 1, f 2, f 3, f 4

},

I 2(f 1) = ( u1, u2), I 2(f 2) = ( u2, u3), I 2(f 3) = ( u3, u1), I

i.e., those graphs shown in Fig.1 .4.2.

u1

v2v3

e1

e2

e3

e4

v1

u3

f 1

f 4



22 Chap.1 Com

for ∀e∈E 1. Therefore, φ is an isomorphism between G1 and Gare isomorphic.

If G1 = G2 = G, an isomorphism between G1 and G2 is calledof G. All automorphisms of a graph G form a group under the cotion, i.e., φθ(x) = φ(θ(x)), where x∈E (G) V (G). We denote th

group by Aut G.For a simple graph G of n vertices, it can be veried that

symmetry group action on n vertices of G. But for non-simple grapmore complex. For example, the automorphism groups of graphsin Fig.1.4.3, respectively called complete graphs and bouquets, are Aut B

n= S

n, where m =

|V (K

m)

|and n =

|E (B

n)

|.

K 6 B4

Fig. 1.4.3

1.4.2 Subgraph. A graph H = ( V 1, E 1; I 1) is a subgraph of a gif V 1 ⊆V , E 1 ⊆E and I 1 : E 1 →V 1 ×V 1. We use H ⊂G to

a subgraph of G. For example, graphs G1, G2, G3 are subgraphs Fig.1.4.4.

u1 u2 u1 u2 u1 u2



Sec.1.4 Graphs

induced if H ∼= U for some subset U of V (G). Similarly, F of E (G), the subgraph F induced by F in G is a graphwhose vertex set consists of vertices of G incident with at lsubgraph H of G is edge-induced if H ∼= F for some subsetsubgraphs G1 and G2 are both vertex-induced subgraphs

edge-induced subgraphs {(u1, u4)}, {(u2, u3)}.For a subgraph H of G, if |V (H )| = |V (G)|, then

subgraph of G. In Fig.4.6, the subgraph G3 is a spanning suA complete subgraph of a graph is called a clique, and

spanning subgraph also called a k-factor .

1.4.3 Labeled Graph. A labeled graph on a graph G =θL : V ∪E →L for a label set L, denoted by GL . If θL :then GL is called a vertex labeled graph or an edge labeled gGE , respectively. Otherwise, it is called a vertex-edge labeltwo vertex-edge labeled graphs on K 4 are shown in Fig.1.4.

1

2

3

11

1

4 2

3

2

3 44

1 1 2

2 12

Fig. 1.4.5Two labeled graphs GL1

1 , GL22 are equivalent , denoted by

an isomorphism τ : G1 →G2 such that τθL1 (x) = θL2 τ (x) foWh ll id i l l l b l d h



24 Chap.1 Com

C 2. Hamiltonian graph. A graph G is hamiltonian if it haa hamiltonian circuit containing all vertices of G. Similarly, a pvertices of a graph G is called a hamiltonian path .

C 3. Bouquet and dipole. A graph Bn = ( V b, E b; I b) with V

{e1, e2, · · ·, en}and I b(ei ) = ( O, O) for any integer i, 1 ≤ i ≤n is

n edges. Similarly, a graph D s.l.t = ( V d, E d; I d) is called a dipoleE d = {e1, e2, · · ·, es , es+1 , · · ·, es+ l, es+ l+1 , · · ·, es+ l+ t}and

I d(ei ) =⎧⎪⎪

⎨⎪⎪⎩

(O1, O1), if 1≤ i ≤s,(O1, O2), if s + 1 ≤ i ≤s + l,

(O2, O2), if s + l + 1 ≤ i ≤s + l + t

For example, B3 and D2,3,2 are shown in Fig.1.4.6.

OO1 O2

Fig. 1.4.6

The behavior of bouquets on surfaces fascinated many mathetion. By a combinatorial view, these connected sums of tori, or

sums of projective planes used in topology are just bouquets on face.

C 4. Complete graph. A complete graph K n = ( V c, E c; I c) is a V { } E { 1 i j i j } d I ( )



Sec.1.4 Graphs

consequence, a tree or a forest is a bipartite graph since both oLet G = ( V, E ; I ) be an r-partite graph and V 1, V 2, · ·

subsets. If there is an edge eij ∈E for∀vi∈V i and∀v j ∈V j , such that I (e) = ( vi , v j ), then G is called a complete r -paG = K (|V 1|, |V 2|, · · ·, |V r |). By this denition, a complete g

complete 1-partite graph.

C 6. Regular graph. A graph G is regular of valency k if ρG

These graphs are also called k-regular . A 3-regular graph is ograph .

C 7. Planar graph. A graph is planar if it can be drawnway that edges are disjoint expect possibly for endpoints. Whand edges of a planar graph G from the plane, each remaincalled a face of G. The length of the boundary of a face is planar graphs are shown in Fig.1 .4.7.

tetrahedron cube

Fig. 1.4.7

C 8. Embedded graph. A graph G is embeddable into athere is a one-to-one continuous mapping f : G → Rin sucdisjoint except possibly on endpoints. A embedded graph on



26 Chap.1 Com

A graph consists of k disjoint copies of a graph H , k ≥1 is deAs an example, we nd that

K 6 =5

i=1

S 1.i

for graphs shown in Fig.1 .4.8 following

1

2 34

56

2

34

5

63

45

64

5

6

S 1.5 S 1.4 S 1.3 S 1.2

Fig. 1.4.8

and generally, K n =n − 1

i=1S 1.i . Notice that kG is a multigraph wit

for any integer k, k ≥2 and a simple graph G.

A complement G of a graph G is a graph with vertex set V (G)are adjacent in G if and only if these are not adjacent in G. A jwith G2 is dened by

V (G1 + G2) = V (G1) V (G2),

E (G1 + G2) = E (G1) E (G2)

{(u, v)

|u

∈

V (G1), v

∈and

I (G1 + G2) = I (G1) I (G2) {I (u, v) = ( u, v)|u∈V (G1),



Sec.1.5 Enumeration

§1.5 ENUMERATION

1.5.1 Enumeration Principle. The enumeration problecount and nd closed formula for elements in this set. A funsolving this problem in general is on account of the enumera

for nite sets X and Y , the equality |X | = |Y | holds bijection f : X →Y .

Certainly, if the set Y can be easily countable, then we can elements in X .

1.5.2 Inclusion-exclusion principle. By denition, thesets X and Y are known.

|X ×Y | = |X ||Y |,

|X Y

|=

|X

|+

|Y

| − |X Y

|.

Usually, the rst equality is called the product principle andexclusion principle can be generalized to n sets X 1, X 2, · · ·,Theorem 1.5.1 Let X 1, X 2, · · ·, X n be nite sets. Then

|n

i=1X i | =

n

s=1(−1)s+1

{i1 ,··· ,i s }⊆{1,2,··· ,n } |X i1 X i2

Proof To prove this equality, assume an element x∈n

i 1



28 Chap.1 Com

s1 −

s2

+ · · ·+ ( −1)s ss

= 1 −(1 −1)

times in

n

s=1(−1)

s+1

{i1 ,··· ,i s }⊆{1,2,··· ,n } |X i1 X i2 · · · X

Whence, we get

|n

i=1

X i| =n

s=1

(−1)s

{i1 ,··· ,i s }⊆{1,2,··· ,n }|X i1 X i2 · · ·

by the enumeration principle.

The inclusion-exclusion principle is very useful in dealing problems. For example, an Euler function ϕ is an mapping ϕinteger set Z+ given by

ϕ(n) = |{k∈Z|0 < k ≤n and (k, n) = 1}|,for any integer n ∈Zn , where (k, n ) is the maximum common dAssume all prime divisors in n are p1, p2, · · ·, pl and dene

X i

=

{k

∈

Z

|0 < k

≤n and (k, n ) = p

i},

for any integer i, 1 ≤ i ≤ l. Then by the inclusion-exclusion princi



Sec.1.5 Enumeration

= nl

i=1

(1 − 1 pi

).

1.5.3 Enumerating Mappings. This subsection concention of bijections, injections and surjections from a given senience, dene three sets

Bij (Y X ) = {f ∈Y X |f is an bijection

Inj (Y X ) = {f ∈Y X |f is an injection

Sur (Y X ) = {f ∈Y X |f is an surjection

Then, we immediately get

Theorem 1.5.2 Let X and Y be nite sets. Then

|Bij (Y X )| =0 if |X | = |Y |,|Y |! if |X | = |Y |

and

|Inj (Y X )| =0 if |X | > |Y

|Y |!( |Y |−| X |)! if |X | ≤ |Y

Proof If |X | = |Y |, there are no bijections from X to Ywe only need to consider the case of

|X

|=

|Y

|. Let X =

{x

{y1, y2, · · ·, yn}. For any permutation p on y1, y2, · · ·, yn , thby

x x x



30 Chap.1 Com

surjections from X to Y . Now there are |Y ||X |

ways choosin

Y . Therefore, the number |Inj (Y X )| of surjections from X to Y

|Y |

|X

||X |! = |Y |!

(

|Y

|!

− |X

|!)

.


The situation for |Sur (Y X )| is more complicated than these ca

|Bij (Y X )| and |Inj (Y X )|, which need to apply the inclusion-exclustechniques.

Theorem 1.5.3 Let X and Y be nite sets. Then

|Sur (Y X )| = ( −1)|Y ||Y |

i=0

(−1)i |Y |i

i |X | .

Proof For any sets X = {x1, x2, · · ·, xn}and Y , by the pro

know that

|Y X | = |Y {x1 } ×Y {x2 } ×· · ·×Y {x n }|= |Y {x1 }||Y {x2 }| · · · |Y {xn }| = |Y ||X | .

Now let Φ : Y X

→P (Y ) be a mapping dened by

Φ(f ) = Y f (X ) −Y f (X ).

Notice that f ∈Sur (Y X ) is a surjection if and only if Φ(f ) =



Sec.1.5 Enumeration

Applying the inclusion-exclusion principle, we nd that

|Sur (Y X )| = |Y X \∅= S ⊆Y

X S |

= |Y X | −|Y |

i=1(−1)|S |(|Y | −

=|Y |

i=0

(−1)i

|S |= i

(|Y | −i) |X |

=|Y |

i=0

(

−1)i |Y |

i(

|Y

| −i

= ( −1)|Y ||Y |

i=0

(−1)i |Y |i

The last equality applies the fact |Y |i

= |Y ||Y | −cients.

1.5.4 Enumerating Labeled Graphs. For a given graphcan how many non-equivalent labeled graphs GL be obtainefollowing.

Theorem 1.5.4 Let G be a graph and L a nite labeled set

|L||V (G) |+ |E (G) |

|Aut G|2



32 Chap.1 Com

|L||V (G) |

|Aut G|non-equivalent vertex labeled graphs by labeling θL : V (G) →L

Similarly, for a given vertex labeled graph GV , there are

|L||V (G) |

|Aut G|non-equivalent edge labeled graphs by labeling θL : E (G) →Whence, applying the product principle for enumeration, we nd t

|L

||V (G) |+ |E (G) |

|Aut G|2non-equivalent labeled graphs by labeling θL : V (G)∪E (G) →L

If each element in L appears one times at most, i.e. |θL

∀x∈V (G)∪E (G), then |L| ≥ |V (G)|+ |E (G)| if there exist succase, there are

|L||V (G)|+ |E (G)|

labelings θL : V (G)∪E (G) →L with |θL (x) ∩L| ≤1. Particula

|V (G)

|+

|E (G)

|as usual, then there are (

|V (G)

|+

|E (G)

|)! such

to Theorem 1 .5.4, we know the result following.

Theorem 1.5.5 Let G be a graph and L a nite labeled set wi

|E (G)| Then there are



Sec.1.5 Enumeration

non-equivalent labeled graphs if |L| = |V (G)|+ |E (G)|.For vertex or edge labeled graphs,i.e., |L| = |V (G)| or |L

similar results on the numbers of non-equivalent such labelefollowing.

Corollary 1.5.1 Let G be a graph. Then there are

|V (G)|!|Aut G|

or |E (G)|!|Aut G|

non-equivalent vertex or edge labeled graphs.

There is a closed formula for the number of non-equivaltrees with a given order, shown in the following.

Theorem 1.5.6 Let T be a tree of order p. Then there are

(2 p−1) p− 2( p + 1)!

non-equivalent vertex-edge labeled trees.

Proof Let T be a vertex-edge labeled tree with a label seRemove the pendent vertex having the smallest label a1 andlabel c1. Assume that b1 was the vertex adjacent to a1. p

−1 vertices let a2 be the pendent vertex with the smallest l

adjacent to a2. Remove the edge (a2, b2) with label c2. Repeon the remaining p−2 vertices, and then on p−3 vertices, andafter p −2 steps as only two vertices are left. Then the v

i l d

34



34 Chap.1 Com

3

2

7

61

84

9 5

Fig. 1.5.1

Conversely, given sequences (b1, b2, · · ·, b p− 2) and ( c1, c2, · · ·,bels, a vertex-edge labeled tree of order p can be uniquely constru

First, determine the rst number in 1 , 2, 3, · · ·, 2 p −1 thatin (b1, b2, · · ·, b p− 2), say a1 and dene an edge (a1, b1) with a lab1, c1 from these sequences. Find a smallest number not appearing sequence (b2, c2, · · ·, b p− 2, c p− 2), say a2 and dene an edge (a2, b2

This construction is continued until there are no element left. At thtwo elements remaining in L are connected with the label c p− 1.

For each of the p −2 elements in the sequence (5 −1), we cof numbers in L, thus

(2 p−1) p− 2

( p−2)-tuples. For the remained two vertices and elements in the swe have

p + 1 p−1

2! = ( p + 1)!

choices. Therefore, there are

S 1 6 R k



Sec.1.6 Remarks

Theorem 1.5.6(Cayley, 1889) Let T be a tree of order pnon-equivalent vertex labeled trees.

These enumerating results in Theorems 1 .5.5−1.5.6 caities combining with Theorem 5 .4 and Corollary 1.5.1.

Corollary 1.5.2 Let

T ( p

−1) be a set of trees of order p. T

T ∈T ( p− 1)

1

|Aut T |=

p p− 2

p!

and

T ∈T ( p− 1)

1|Aut T |2

= (2 p−1) p− 2

( p + 1)(2 p−1)!

These equalities are interesting, which present closed fphism groups of trees with given size. The rst equality in Cnoted by Babai in 1974.

§1.6 REMARKS

1.6.1. Combinatorics has made great progress in the 20t

important results found. Essentially, it can be seen as an sets or a branch of algebra with some one’s intuition, suchit is indeed come into being under the logic, namely, a suFor materials in Sections 1 1 1 3 further information and

36 Chap 1 Com



36 Chap.1 Com

torial structure. Further research on multi-poset will enrich oneposets.

1.6.3 These graph families enumerated in Section 4 is not completecommon families or frequently met in papers on graphs. But forded graphs, more words should be added in here. Generally, an

on a topological space Ris a one-to-one continuous mapping f :way that edges are disjoint except possibly on endpoints, namely, embedded in a topological space [Gr u1]. In last century, many recentrated on the case of Rbeing a surface, i.e., a closed 2-manterminology embedded graph is usually means a graph embedded o

a general topological space. For this spacial case, more and more tcombinatorics are applied, for example, [GrT1], [Whi1] and [Maowith algebra, particularly, automorphism groups, [Liu1]-[Liu3] ualgebra, algorithm, mathematical analysis, particularly, functional[MoT1] adopts combinatorial topology. Certainly, there are manyin this eld. Beyond embedded graphs on surfaces, few results publications for embedded graphs in a topological space, not these

1.6.4 The identity of automorphism groups of trees

T ∈T ( p− 1)

1

|Aut T |2=

(2 p−1) p− 2( p + 1)!(2 p−1)!

in Corollary 1.5.2 is a new identity. Generally, two different wayon a given conguration induce a combinatorial identity. In [MaLan identity of automorphism groups of trees different from these in

Sec 1 6 Remarks



Sec.1.6 Remarks

is only benecial for pure or classical combinatorics, not the eor sciences for its lack of metrics. The goal of combinatorics counterpart in mathematics, not just these results only with pimportance. For its contribution to the entire science, a good metrics ignored in classical combinatorics to construct the

torics suggested by the author in [Mao1]. The reference [Maowith Smarandache multi-spaces. In fact, the material in theon mathematical combinatorics, particularly on combinatoriand its application, i.e., combinatorial elds in theoretical ph



CHAPTER 2.

Fundamental of

Mathematical Combinatorics

One increasingly realizes that our world is not an individual but a or combinatorial one, which enables modern sciences overlap and

i.e., with a combinatorial structure. To be consistency with thedevelopment, the mathematics should be also combinatorial, not classical combinatorics without metrics, but the mathematical combresulting in the combinatorial conjecture for mathematics, i.e., matscience can be reconstructed from or made by combinatorialization by the author in 2005. The importance of this conjecture is not inan open problem, but in its role for advancing mathematics. For intrmore readers known this heartening combinatorial notion for mathesciences, this chapter introduces the combinatorial algebraic theor

Sec.2.1 Combinatorial Systems



y

§2.1 COMBINATORIAL SYSTEMS

2.1.1 Proposition in Logic. The multi-laterality of our Wsystems to be its best candidate model for ones cognition onalso included in a well-known Chinese ancient book TAO TLAO ZI . In this book we can nd many sentences for cognsuch as those of the following ([Luj1]-[Luj2],[Sim1]).SENTENCE 1. All things that we can acknowledge is detears, or nose, or tongue, or body or passions, i.e., these six shown in Fig.2.1.1.

known partby ones six organs

unknown

unknounknown

unknown

Fig. 2.1.1SENTENCE 2. The Tao gives birth to One. One givgives birth to Three. Three gives birth to all things. All thi

the female and stand facing the male. When male and femachieve harmony. Shown in Fig.2.1.2.

¹ +1

40 Chap.2 Fundamental of Mathemat



SENTENCE 3. Mankind follows the earth. Earth follows tuniverse follows the Tao. The Tao follows only itself. Such aFig.2.1.3.

mankind earth universe TAO

Fig. 2.1.3

SENTENCE 4. Have and Not have exist jointly ahead of theand the sky. This means that any thing have two sides. One is the p

is the negative. We can not say a thing existing or not just by our siits existence independent on our living.

What can we learn from these words? All these sentences meis a multi-one. For characterizing its behavior, We should construcmodel for the WORLD, also called parallel universes ([Mao3], [Teg

shown in Fig.2.1.4.

known part now

unknown unknown




proposition are shown in Fig.2 .1.5.

p anti-p¹

p

non-p

non-p

non-p

Fig.2.1.5

For a given proposition, what can we say it is true or faits non-proposition jointly exist in the world. Its truth or fals

by logic inference, independent on one knowing it or not.A norm inference is called implication. An implication

is a proposition that is false when p is true but q false andare three propositions related with p →q, namely, q → p, ¬called the converse , contrapositive and inverse of p →q. Two

equivalent if they have the same truth value. It can be shownimplication and its contrapositive are equivalent . This facmathematical proofs, i.e., we can either prove the propositionthe proof of p →q, not the both.

2.1.2 Mathematical System. A rule on a set Σ is a ma

Σ ×Σ · · ·×Σn

→Σ

for some integers n A mathematical system is a pair (Σ




for mathematical systems are shown in the following.Example 2.1.1 A group (G;◦) in classical algebra is a mathematicawhere Σ G = G and

RG = {RG1 ; RG

2 , RG3 },

with

RG1 : (x ◦y) ◦z = x ◦ (y ◦z) for∀x,y,z ∈G;

RG2 : there is an element 1G∈G such that x ◦1G = x for∀x

RG3 : for∀x∈G, there is an element y, y∈G, such that x ◦

Example 2.1.2 A ring (R; + , ◦) with two binary closed operationmathematical system (Σ; R), where Σ = R and R= {R1; R2, R

R1: x + y, x ◦y∈R for∀x, y∈R;R2: (R; +) is a commutative group, i.e., x + y = y + x for∀R3: (R;◦) is a semigroup;R4: x

◦(y + z) = x

◦y + x

◦z and (x + y)

◦z = x

◦z + y

◦Example 2.1.3 a Euclidean geometry on the plane R 2 is a a ma(Σ E ;RE ), where Σ E = {points and lines on R 2}and RE = {HilbEuclidean geometry }.

A mathematical (Σ; R) can be constructed dependent on the

R. The former requires each rule in Rclosed in Σ. But the lR(a,b, · · ·, c) in the nal set Σ, which means that Σ maybe an exΣ. In this case, we say Σ is generated by Σ under rules R, denot

Combining mathematical systems with the view of LAO Z




Denition 2.1.2 For an integer m ≥ 2, let (Σ 1;R1), (Σ 2

m mathematical systems different two by two. A Smarandach(Σ; R) with

Σ =m

i=1

Σ i , and R=m

i=1Ri .

Certainly, we can construct Smarandache systems by apmulti-spaces, particularly, Smarandache geometries appeared

2.1.3 Combinatorial System. These Smarandache systDenition 2.1.1 consider the behavior of a proposition and

the same set Σ without distinguishing the guises of these nonthere are many appearing ways for non-propositions of a pdescribing their behavior, we need combinatorial systems.

Denition 2.1.3 A combinatorial system C G is a union of(Σ 1;R1),(Σ 2;R2), · · ·, (Σ m ;Rm ) for an integer m, i.e.,

C G = (m

i=1

Σ i ;m

i=1Ri )

with an underlying connected graph structure G, where

V (G) = {Σ 1, Σ 2, · · ·, Σ m},

E (G) = {(Σ i , Σ j ) | Σ i Σ j = ∅, 1 ≤ i, j ≤Unless its combinatorial structure G, these cardinalitie

coupling constants in a combinatorial system C G also det




|Σ i (aR(1)i b) = |Σ i (a) |Σ i (R

(1)i ) |Σ i (b)

for∀a, b∈Σ (1)i , 1 ≤ i ≤m, where |Σ i denotes the constraint map

mathematical system (Σ i , Ri). Further more, if : C (1)G →C (2)

G ithen we say these C (1)

G and C (2)G are isomorphic with an isomor

them.A homomorphism : C (1)

G →C (2)G naturally induces a mapp

graph G1 and G2 by

|G : V (G1) → (V (G1))⊂V (G2) and

|G : (Σ i , Σ j )∈E (G1) →( (Σ i), (Σ j ))∈E (G2), 1 ≤With these notations, a criterion for isomorphic combinatorial systein the following.

Theorem 2.1.1 Two combinatorial systems C (1)G and C (2)

G are

only if there is a 1 −1 mapping : C (1)G →C (2)

G such that

(i) |Σ (1)i

is an isomorphism and |Σ (1)i

(x) = |Σ (1)j

(x) for ∀xi, j ≤m;

(ii ) |G : G1 →G2 is an isomorphism.

Proof If :C (1)

G →C (2)

G is an isomorphism, considering thpings of on the mathematical system (Σ i , Ri) for an integer i,graph G(1)

1 , then we nd isomorphisms |Σ (1)i

and |G .Conversely, if these isomorphism | (1) , 1 ≤ i ≤ m and




For understanding well the multiple behavior of world, ashould be constructed. Then what is its relation with classences? What is its developing way for mathematical scienceof combinatorial notion in Chapter 5 of [Mao1], then formallconjecture for mathematics in [Mao4] and [Mao10], the lateConference on Combinatorics and Graph Theory of China i

Combinatorial Conjecture Any mathematical system (Σsystem C G (lij , 1 ≤ i, j ≤m).

This conjecture is not just an open problem, but more liwhich opens a entirely way for advancing the modern mathe

physics. In fact, it is an extending of TAO TEH KING , Smcombinatorics, but with more delicateness.

Here, we need further clarication for this conjecture. Ina combinatorial notion on mathematical objects following fo

(i) There is a combinatorial structure and nite rulesematical system, which means one can make combinatorialimathematical subjects.

(ii ) One can generalizes a classical mathematical systemnotion such that it is a particular case in this generalization.

(iii ) One can make one combination of different branchnd new results after then.

(iv) One can understand our WORLD by this combinatcombinatorial models for it and then nd its behavior for ex




§2.2 ALGEBRAIC SYSTEMS

2.2.1 Algebraic System. Let A be a set and ◦ an operatA ×A →A , i.e., closed then we call A an algebraic system un

◦, denoted by ( A ;◦). For example, let A = {1, 2, 3}. Dene opA by following tables.

×1 1 2 31 1 2 32 2 3 13 3 1 2

×2 1 2 31 1 2 2 3 1 3 2 3

table 2.2.1

Then we get two algebraic systems ( A ;×1) and (A ;×2). Notice thsystem (A ;◦), we can get an unique element a ◦b∈

A for∀a, b

2.2.2 Associative and Commutative Law. We introduce th

commutative laws in the following denition.

Denition 2.2.1 An algebraic system (A ;◦) is associative if

(a ◦b) ◦c = a ◦ (b◦c)

for

∀

a,b,c

∈

A .

Denition 2.2.2 An algebraic system (A ;◦) is commutative if

a b = b a

Sec.2.2 Algebraic Systems



(· · ·((a1 ◦a2) ◦a3) ◦ · · ·) ◦an .

If n = 3, the claim is true by denition. Assume theintegers n ≤k. We consider the case of n = k + 1. By deany calculating order must be a result of two elements, i.

=1

◦2

.

Apply the inductive assumption, we can assume that

1

= (

· · ·((a1

◦a2)

◦a3)

◦ · · ·)

◦a l

and

2

= ( · · ·((a l+1 ◦a l+2 ) ◦a l+3 ) ◦ · · ·) ◦ak

Therefore, we get that

=1

◦2

= ( · · ·(a1 ◦a2) ◦ · · ·) ◦a l ◦ (· · ·(a l+1 ◦a l+2 ) ◦ ·= (

· · ·(a1

◦a2)

◦ · · ·)

◦a l

◦((

· · ·(a l+1

◦a l+2 )

◦= (( · · ·(a1 ◦a2) ◦ · · ·) ◦a l ◦ (· · ·(a l+1 ◦a l+2 ) ◦= ( · · ·((a1 ◦a2) ◦a3) ◦ · · ·) ◦ak+1




aπ (1) ◦aπ (2) ◦ · · · ◦aπ (l) = a1 ◦a2 ◦ · · · ◦a l .

Not loss of generality, let π(k) = n. Then we know that

aπ (1) ◦aπ (2) ◦ · · · ◦aπ (n ) = ( aπ (1) ◦aπ (2) ◦ · · · ◦aπ (k− 1) )

◦an ◦ (aπ (k+1) ◦aπ (k+2) ◦ · ·= ( aπ (1) ◦aπ (2) ◦ · · · ◦aπ (k− 1) )

◦((aπ (k+1) ◦aπ (k+2) ◦ · · · ◦aπ

= (( aπ (1) ◦aπ (2) ◦ · · · ◦aπ (k− 1) )

◦(aπ (k+1)

◦aπ (k+2)

◦ · · · ◦aπ (

= a1 ◦a2 ◦ · · · ◦an

by the inductive assumption.

Let (A ;◦) be an algebraic system. If there exists an elemethat

1l◦ ◦a = a or a ◦1r

◦ = a

for∀a∈A , then 1 l

◦ (1r◦ ) is called a left unit (or right unit ) in (A

exist simultaneously, then there must be

1l◦ = 1 l

◦ ◦1r◦ = 1 r

◦ = 1 ◦ ,

i.e., a unit 1◦ in (A ;◦). For example, the algebraic system ( A ;previous examples is a such algebraic system, but ( A ;×2) only1× 2 = 1.




called nite ( or innite ) if |A | is nite ( or innite). Forpermutations Π( X ) under operations ×1, composition on a groups (A ;×1) and Sym(X ) respectively.

2.2.4 Isomorphism of Systems. Two algebraic systemare called homomorphic if there exists a mapping ς : A 1 →A

ς (a) ◦2 ς (b) for ∀a, b ∈A 1. If this mapping is a bijection

systems are called isomorphic . In the case of A 1 = A 2 = A

isomorphism between ( A 1;◦1) and (A 2;◦2) is called an auto

Theorem 2.2.3 Let (A ;◦) be an algebraic system. Then(A ;◦) form a group under the composition operation, denote

Proof For two automorphisms ς 1 and ς 2 on (A ;◦), It is

ς 1ς 2(a ◦b) = ς 1ς 2(a) ◦ς 1ς 2(b)

for∀a, b∈A by denition, i.e., Aut (A ;◦) is an algebraic s

morphism 1f ix by 1f ix (a) = a and an automorphism ς − 1 by

for∀a, b∈A . Then 1f ix is the unit and ς

− 1is the inverse ele

By denition, Aut (A ;◦) is a group under the composition o

2.2.5 Homomorphism Theorem. Now let (A ;◦) be aB⊂

A , if (B ;◦) is still an algebraic system, then we csystem of (A ;◦), denoted by B

≺A . Similarly, an algebra

a subgroup if it is group itself.Let (A ;◦) be an algebraic system and B

≺A . For

a ◦B of B in A by




(a ◦B ) • (b◦B ) = ( a ◦b) ◦B ,

then (S ;•) is an associative algebraic system, called a quotient syParticularly, if there is a representation R whose each element(A ;◦) with unit 1A , then (S ;•) is a group, called a quotient grou

Proof For ( i), notice that if

(a ◦B ) ∩(b◦B ) = ∅for a, b ∈

A , then there are elements c1, c2 ∈B such that a

assumption, ( B ;◦) is a subgroup of (A ;◦), we know that there

element c− 11 ∈B , i.e., a = b◦c2 ◦c− 11 . Therefore, we get that

a ◦B = ( b◦c2 ◦c− 11 ) ◦B

= {(b◦c2 ◦c− 11 ) ◦c|∀c∈B }

= {b◦c|∀c∈B

}= b◦B

by the associative law and ( B ;◦) is a group gain, i.e., (a ◦ B )a ◦B = b◦B .

By denition of • on S and ( i), we know that ( S ;•) is an

For ∀a,b,c∈A , by the associative laws in ( A ;◦), we nd that

((a B ) (b B )) (c B ) = (( a b) B ) (c B )




Corollary 2.2.2(Lagrange theorem) For a subgroup (B ;◦)

|B | | |A |.Proof Since a ◦c1 = a ◦c2 implies that c1 = c2 in this c

|a ◦B | = |B |for∀a∈

A . Applying Theorem 2 .2.4(i), we nd that

|A | =r∈R

|r ◦B | = |R||B |,for a representation R, i.e.,

|B

| | |A

|.

Although the operation • in S is introduced by the opbe •= ◦. Now if •= ◦, i.e.,

(a ◦B ) ◦ (b◦B ) = ( a ◦b) ◦B ,

the subgroup (B

;◦) is called a normal subgroup of (B

;◦), this case, if there exist inverses of a, b, we know that

B ◦b◦B = b◦B

by product a− 1 from the left on both side of (2.2.1). Now sin

we get that

b− 1 ◦B ◦b = B ,

hi h i h ll d i i f l b f




− 1(a2) = {a1∈A 1| (a1) = a2}.

Particularly, if a2 = 1 A 2 , the inverse set − 1(1A 2 ) is important in athe kernel of and denoted by Ker( ), which is a normal subgrois associative and each element in Ker( ) has inverse element in

by denition, for ∀a,b,c∈A 1, we know that

(1) (a ◦b) ◦c = a ◦(b◦c)∈Ker( ) for ((a ◦b) ◦c) = (a

(2) 1A 2 ∈Ker( ) for (1A 1 ) = 1 A 2 ;

(3) a− 1∈Ker( ) for ∀a ∈Ker( ) if a− 1 exists in (A 1;◦

− 1(a) = 1 A 2 ;

(4) a ◦Ker( ) = Ker( ) ◦a for

(a ◦Ker( )) = (Ker( ) ◦a) = − 1( (a))

by denition. Whence, Ker( ) is a normal subgroup of (A 1;◦1).

Theorem 2.2.5 Let : A 1 →A 2 be an onto homomorphismsystems (A 1;◦1) to (A 2;◦2) with units 1A 1 , 1A 2 . Then

A 1/ Ker( ) ∼= (A 2;◦2)

if each element of Ker( ) has an inverse in (A 1;◦1).

Proof We have known that Ker( ) is a subgroup of (A 1;

◦1). W

is a quotient system. Dene a mapping ς : A 1/ Ker( ) →A 2 by

ς (a ◦1 Ker( )) = (a).

Sec.2.3 Multi-Operation Systems



= (a) ◦2 (b),

i.e., ς is an isomorphism from A 1/ Ker( ) to (A 2;◦2).

Corollary 2.2.3 Let : A 1 →A 2 be an onto homomorphisto (A 2;◦2). Then

A 1/ Ker( ) ∼= (A 2;◦2).

§2.3 MULTI-OPERATION SYSTEMS

2.3.1 Multi-Operation System. A multi-operation sywith a set H and an operation set

O = {◦i | 1 ≤ i ≤ l}on H such that each pair ( H ;◦i) is an algebraic system. Wl-operation system on H .

A multi-operation system ( H ; O) is associative if for∀athere is

(a ◦1 b) ◦2 c = a ◦1 (b◦2 c).

Such a system is called an associative multi-operation system

Let (H , O) be a multi-operation system and G ⊂

H , Qa multi-operation system, we call ( G ; Q) a multi-operation denoted by ( G ; Q)≺(H , O). In those of subsystems, the (




Then the set

Q = {a ◦G |a∈R, ◦∈P ⊂O}is called a quotient set of G in H with a representation pair (H G |(R, P ) . Similar to Theorem 2 .4, we get the following result.

2.3.2 Isomorphism of Multi-Systems. Two multi-operation and ( H 2; O2) are called homomorphic if there is a mapping ωω : O1 →O2 such that for a1, b1 ∈

H 1 and ◦1 ∈O1, there ex

◦2 = ω(◦1)∈O2 enables that

ω(a1 ◦1 b1) = ω(a1) ◦2 ω(b1).

Similarly, if ω is a bijection, (H 1; O1) and (H 2; O2) are called iH 1 = H 2 = H , ω is called an automorphism on H .

Theorem 2.3.1 Let (H , O) be an associative multi-operation s1

◦for

∀◦∈O and G

⊂

H .

(i) If G is closed for operations in O and for ∀a ∈G , ◦

an inverse element a− 1◦ in (G ;◦), then there is a representation

that the quotient set H G |(R, P ) is a partition of H , i.e., for a, b∈

(a ◦1 G ) ∩(b◦2 G ) = ∅or a ◦1 G = b◦2 G .

(ii ) For ∀◦∈O, dene an operation ◦ on H

G |(R, P ) by

(a ◦1 G ) ◦ (b◦2 G ) = ( a ◦b) ◦1 G .




= b◦2 (g2 ◦1 c− 11 ◦1

G ) = b◦2

by the associative law. This implies that H G |(R, P ) is a partiti

Notice that H G |(R, P ) is closed under operations in P by

operation system. For ∀a,b,c ∈R and operations ◦1, ◦2,that

((a ◦1 G ) ◦1 (b◦2 G )) ◦2 (c ◦3 G ) = (( a ◦1 b) ◦1 G

= (( a ◦1 b) ◦2 c

and

(a ◦1 G ) ◦1 ((b◦2 G ) ◦2 (c ◦3 G )) = ( a ◦1 G ) ◦1 (

= ( a ◦1 (b◦2 c)

by denition. Since ( H , O) is associative, we have (a

◦1 b)

◦2 c

we get that

((a ◦1 G ) ◦1 (b◦2 G )) ◦2 (c ◦3 G ) = ( a ◦1 G ) ◦1 ((b◦2

i.e., ( H G

|(R, P ) ; O) is an associative multi-operation system.

If any element in R has an inverse in (H ;◦ ), then weand a− 1 ◦ G is the inverse element of a ◦ G in the system (is a group again.




inverse x− 1

for ∀x ∈( I (O2) in (( I (O2); ◦−

). Then there are rep(R1, P 1) and (R2, P 2), where P 1⊂O, P 2⊂O2 such that

(H 1; O1)(Kerω; O1) |(R 1 ,P 1 ) ∼=

(H 2; O2)( I (O2); O2) |(R 2 ,P 2 )

if each element of Kerω has an inverse in (H 1;

◦) for

◦∈O1.

Proof Notice that Kerω is an associative subsystem of ( H

∀k1, k2∈Kerω and ∀◦∈O1, there is an operation ◦−∈O2 such

ω(k1 ◦k2) = ω(k1) ◦− ω(k2)∈ I (O2)

since I (O2) is an algebraic system. Whence, Kerω is an associa(H 1; O1). By assumption, for any operation ◦ ∈O1 each elementinverse a− 1 in (H 1;◦). Let ω : (H 1;◦) →(H 2;◦− ). We know that

ω(a ◦a− 1) = ω(a) ◦− ω(a− 1) = 1 ◦ − ,

i.e., ω(a− 1

) = ω(a)− 1

in (H

2;◦−

). Because I (O2) is an algebrainverse x− 1 for ∀x∈ I (O2) in (( I (O2); ◦− ), we nd that ω(a− 1)a− 1∈Kerω.Dene a mapping σ : (H 1 ;O1 )

(Ker ω;O1 ) |(R 1 ,P 1 ) → (H 2 ;O2 )( I (O2 );O2 ) |(R 2 ,P 2 ) by

σ(a

◦Kerω) = σ(a)

◦−

I (O2)

for∀a∈R1, ◦∈P 1, where ω : (H 1;◦) →(H 2;◦− ). We prove σ iNotice that σ is onto by that ω is an onto homomorphism. Nob K ( ) f b ∈R d ∈P h ( ) − (O )



a ◦ (b1 ◦1 b2 ◦2 · · · ◦k bk+1 ) = ( a ◦b1) ◦1 (a ◦b2) ◦2 · · · ◦k− 1 (a ◦= ( a ◦b1) ◦1 (a ◦b2) ◦2 · · · ◦k− 1 (a ◦

by the inductive assumption. Therefore,

a ◦ (b1 ◦1 b2 ◦2 · · · ◦n − 1 bn ) = ( a ◦b1) ◦1 (a ◦b2) ◦2 · · · ◦n −

is hold for any integer n ≥2. Similarly, we can also prove that

(b1 ◦1 b2 ◦2 · · · ◦n − 1 bn ) ◦a = ( b1 ◦a) ◦1 (b2 ◦a) ◦2 · · · ◦n −

2.3.4 Multi-Group and Multi-Ring. An associative multi-(H ;O1 → O2) is said to be a multi-group if (H ;◦) is a group a multi-ring (or multi-eld ) if O1 = {·i|1 ≤ i ≤ l}, O2 = {+ i|1 ≤(or multi-eld) ( H ; + i , ·i) for 1 ≤ i ≤ l. We call them l-group,abbreviation. It is obvious that a multi-group is a group if |O1∪or eld if |O1| = |O2| = 1 in classical algebra. Likewise, We also dof a l-ring (H ;O1 → O2) by 1·i and 0+ i in the ring ( H ; + i , ·i∀a∈

H , by these distribute laws we nd that

a ·i b = a ·i (b + i 0+ i ) = a ·i b + i a ·i 0+ i ,

b

·i a = ( b + i 0+ i )

·i a = b

·i a + i 0+ i

·i a

for∀b∈H . Whence,

a ·i 0+ i = 0 + i and 0+ i ·i a = 0 + i .




(i) multi-subgroup if and only if for ∀a, b∈H, ◦∈O1(ii ) multi-subring if and only if for ∀a, b ∈ H, ·i ∈b, a + i b− 1

+ i ∈ H, particularly, a multi-eld if a ·i b− 1·i , a + i

{·i|1 ≤ i ≤ l}, O2 = {+ i|1 ≤ i ≤ l}.

Proof The necessity of conditions ( i) and ( ii ) is obvious

sufficiency.For ( i), we only need to prove that ( H;◦) is a group

fact, it is associative by the denition of multi-groups. For1◦ = a ◦a− 1

◦ ∈Hand 1◦ ◦a− 1◦ ∈H. Whence, (H;◦) is a gro

Similarly for (ii ), the conditions a ·i b, a+ i b− 1+ i ∈Himply

and closed in operation

·i ∈ O1. These associative or distrib

(H ; + i , ·i) being a ring for any integer i, 1 ≤ i ≤ l. Particulathat ( H; ·i) is also a group. Whence, (H ; + i , ·i) is a eld forin this case.

A multi-ring ( H ;O1 → O2) with O1 = {·i|1 ≤ i ≤ l}is integral if for

∀

a, b

∈

H and an integer i, 1

≤i

≤l, a

and a ◦i b = 0 + i implies that a = 0 + i or b = 0 + i . If l = the integral ring by denition. For the case of multi-rings wintegral multi-ring is nothing but a multi-eld. See the next r

Theorem 2.3.5 A nitely integral multi-ring is a multi-eld

Proof Let (H ;O1 → O2) be a nitely integral multi-ringwhere O1 = {·i|1 ≤ i ≤ l}, O2 = {+ i|1 ≤ i ≤ l}. For any intean element a∈

H and a = 0 + i . Then




in (H

; ·i), which implies it is a commutative group. Therefore, (H

for any integer i, 1 ≤ i ≤ l.

Corollary 2.3.3 Any nitely integral domain is a eld.

2.3.5 Multi-Ideal. Let (H ;O11 → O12), (H ;O2

1 → O22) be

Ok1 =

{·ki

|1

≤i

≤lk

},

Ok2 =

{+ k

i

|1

≤i

≤lk

}for k = 1 , 2 and :

(H ;O21 → O22) a homomorphism. Dene a zero kernel Ker of

Ker0 = {a∈H | (a) = 0 + 2

i, 1 ≤ i ≤ l2}.

Then, for ∀h∈H and a∈Ker0 , (a·1i h) = 0 + i (·i )h = 0 + i ,

Similarly, h ·i a∈Ker0 . These properties imply the conception ofmulti-ring introduced following.

Choose a subset I ⊂H . For ∀h∈H , a∈ I , if there are

h ◦i a∈ I and a ◦i h∈H,then

I is said a multi-ideal . Previous discussion shows that the zer

a homomorphism on a multi-ring is a multi-ideal. Now let I b(H ;O1 → O2). According to Corollary 2.3.1, we know that there pair (R2, P 2) such that

I = {a + i I |a∈R2, + i∈P 2}is a commutative multi-group. By the distributive laws, we nd th

(a + I) (b+ I) = a b + a I+ Ib + I

Sec.2.4 Multi-Modules



Therefore, we conclude that any multi-ideal is a zero keron a multi-ring . The following result is a special case of Th

Theorem 2.3.6 Let (H 1;O11 → O12) and (H 2;O2

1 →ω : (H 1;O1

2) →(H 2;O22 ) be an onto homomorphism with

operation system, where I (O22) denotes all units in (H 2;O2

2

resentation pairs (R1, P 1), (R2, P 2) such that

(H : I )|(R 1 ,P 1 ) ∼=(H 2;O2

1 → O22)( I (O2

2); O22) |(R 2 ,

Particularly, if ( H 2;O21 → O22) is a ring, we get an inter

Corollary 2.3.4 Let (H ;O1 → O2) be a multi-ring, (R(H ;O2) →(R; +) be an onto homomorphism. Then there pair (R, P ) such that

(H : I )|(R, P ) ∼= (R; + , ·).

§2.4 MULTI-MODULES

2.4.1 Multi-Module. There multi-modules are generaliin linear algebra by applying results in last section. Let OO1 = {·i|1 ≤ i ≤ m} and O2 = {+ i|1 ≤ i ≤ m}be a commutative m-group with units 0 + i and ( R ;O1 → unit 1 · for ∀· ∈ O1. For any integer i, 1 ≤ i ≤ m, de


M R



m = 1, It is obvious that Mod (M

(O) :R

(O1 → O2)) is a moif ( R ;O1 → O2) is a eld, then Mod (M (O) : R (O1 → O2)) iclassical algebra.

For any integer k, a i∈ R and xi∈

M , where 1 ≤ i, k ≤s, eare hold by induction on the denition of m-modules.

a ×k (x1 + k x2 + k · · ·+ k xs ) = a ×k x1 + k a ×k x2 + k · · ·+(a1+ ka2+ k · · ·+ kas ) ×k x = a1 ×k x + k a2 ×k x + k · · ·+ k

(a1 ·k a2 ·k · · · ·k as ) ×k x = a1 ×k (a2 ×k · · ·×(as ×k x

and

1·i 1 ×i1 (1·i 2 ×i2 · · ·×is − 1 (1·i s ×is x) · · ·) = xfor integers i1, i2, · · ·, is∈ {1, 2, · · ·, m}.

Notice that for ∀a, x∈M , 1 ≤ i ≤m,

a ×i x = a ×i (x + i 0+ i ) = a ×i x + i a ×i 0+ i ,

we nd that a ×i 0+ i = 0 + i . Similarly, 0 + i ×i a = 0 + i . Applying tthat

a ×i x + i a−+ i ×i x = ( a+ ia−

+ i) ×i x = 0 + i ×i x = 0

and

a ×i x + i a ×i x−+ i = a ×i (x + i x−

+ i ) = a ×i 0+ i = 0

We know that


(i) ι(x + y) = ι(x) + ι(y) for x y M where ι(+



(i) ι(x +iy) = ι(x) + ι(y) for

∀

x, y

∈

M 1, where ι(+

(ii ) ι(a ×i x) = a ×i ι(x) for∀x∈M 1.

If ι is a bijection, these modules Mod (M 1(O1) : R 1(O11

R 2(O21 → O22)) are said to be isomorphic , denoted by

Mod (M 1(

O1) : R 1(

O11

→ O12))∼= Mod (M 2(

O2) : R

Let Mod (M (O) : R (O1 → O2)) be an m-module. (N ;O) of (M ;O), if for any integer i, 1 ≤ i ≤ m, a ×i xx ∈

N , then by denition it is itself an m-module, calledMod (M (O) : R (O1 → O2)).

Now if Mod (N (

O) : R (

O1

→ O2)) is a multi-submo

R (O1 → O2)), by Theorem 2 .3.2, we can get a quotient mua representation pair ( R, P ) under operations

(a + i N ) + ( b+ j N ) = ( a + b) + i N

for

∀

a, b

∈

R, +

∈ O. For convenience, we denote elements

x(i) . For an integer i, 1 ≤ i ≤m and ∀a∈ R , dene

a ×i x(i) = ( a ×i x)(i) .

Then it can be shown immediately that

(i) a ×i (x(i)

+ i y(i)

) = a ×i x(i)

+ i a ×i y(i)

;(ii ) (a+ ib) ×i x(i) = a ×i x(i) + i b×i x(i) ;

(iii ) (a ·i b) ×i x(i) = a ×i (b×i x(i) );


i n and ι : Mod (M ( ) : R ( 1 1)) Mod (M ( )



i

≤n

}and ι : Mod (M

1(

O1) : R

1(

O1 → O2))

→Mod (M

2(

O2)

be a onto homomorphism with ( I (O2); O2) a multi-group, whereunits in the commutative multi-group (M 2;O2). Then there expairs (R1, P 1), (R2, P 2) such that

Mod (M / N )

|(R 1 ,P 1 )

∼

= Mod (M 2(

O2)/

I (

O2))

|(R 2

where N = Ker ι is the kernel of ι. Particularly, if ( I (O2); O|I (O2)| = 1 , then

Mod (M / N )|(R 1 ,P 1 ) ∼= Mod (M 2(O2) : R 2(O21 → O22)

Proof Notice that ( I (O2); O2) is a commutative multi-group. construct a quotient module Mod (M 2(O2)/ I (O2)). Applying Tnd that

Mod (M / N )|(R 1 ,P 1 ) ∼= Mod (M 2(O2)/ I (O2))|(R 2

Notice that Mod (M 2(O2)/ I (O2)) = Mod (M 2(O2) : R 2(Ocase of |I (O2)| = 1. We get the isomorphism as desired.

Corollary 2.4.1 Let Mod (M (O) : R (O1 → O2)) be an m-

{+ i | 1 ≤ i ≤m}, O1 = {·i|1 ≤ i ≤m}, O2 = {+ i|1 ≤ i ≤m},ring (R; + ,

·) and ι : Mod (M

1(

O1) : R

1(

O1

1 → O1

2))

→M a on

with Kerι = N . Then there exists a representation pair (R , P ) s

M d (M / N )| M


M



Let S ⊂

M with | S | = n. Dene its linearly spanning set R (O1 → O2)) to be

S | R = {m

i=1

n

j =1

α ij ×i xij | α ij ∈ R , x ij

where

m

i=1

n

j =1

a ij ×ij xi = a11 ×1 x11 + 1 · · ·+ 1 a1n ×+ (1) a21 ×2 x21 + 2 · · ·+ 2 a

+ (2)

· · · · · · · · · · · · · · ·am 1 ×m xm 1 + m · · ·+ m am

with + (1) , + (2) , + (3)∈ Oand particularly, if + 1 = + 2 = ·

bym

i=1xi as usual. It can be checked easily that S | R i

Mod (M (

O) : R (

O1

→ O2)), call it generated by S in M

O2)). If S is nite, we also say that S | R is nitely gen

S = {x}, then S | R is called a cyclic multi-submodule of M

O2)), denoted by R x. Notice that

R x =

{

m

i=1

a i

×i x

|a i

∈

R

}by denition. For any nite set S , if for any integer s, 1 ≤s

m s i


Dene operations



Dene operations

(x1, x2, · · ·, xn ) + i (y1, y2, · · ·, yn ) = ( x1+ iy1, x2+ iy2, · · ·and

a

×i (x1, x2,

· · ·, xn ) = ( a

·i x1, a

·i x2,

· · ·, a

·i xn

for∀a∈ R and integers 1 ≤ i ≤m. Then it can be immediately

is a multi-module Mod ( R (n ) : R (O1 → O2)). We construct a bamulti-module in the following.

For any integer k, 1 ≤k ≤n, let

e1 = (1 ·k , 0+ k , · · ·, 0+ k );e2 = (0 + k , 1·k , · · ·, 0+ k );

· · · · · · · · · · · · · · · · · ·;

en = (0 + k , · · ·, 0+ k , 1·k ).

Notice that(x1, x2, · · ·, xn ) = x1 ×k e1 + k x2 ×k e2 + k · · ·+ k xn ×

We nd that each element in R (n ) is generated by e1, e2, · · ·, en .

(x1, x2, · · ·, xn ) = (0 + k , 0+ k , · · ·, 0+ k )

implies that xi = 0 + k for any integer i, 1 ≤ i ≤ n. Whence, {basis of Mod ( R (n ) : R (O1 → O2)).

Theorem 2 4 2 Let Mod (M (O) : R (O1 O2)) = S|R be a

Sec.2.5 Actions of Multi-Graphs

Whence, ϑ is a homomorphism. Notice that it is also 1



, p

−that ϑ is an isomorphism between Mod (M (O) : R (O1 → R (O1 → O2)).

§2.5 ACTIONS OF MULTI-GROUPS

2.5.1 Construction of Permutation Multi-Group.be a nite set. As dened in Subsection 1.3.1, a compospermutations

τ =x1 x2

· · ·xn

y1 y2 · · · yn , ,

and

ς =y1 y2 · · · yn

z1 z2 · · · zn ,,

are dened to be

σ =x1 x2 · · · xn

y1 y2 · · · yn ,y1 y2 · · · yn

z1 z2 · · · zn ,=

xz

As we have pointed out in Section 2 .2.3, all permutatiounder the composition operation.

For ∀ p∈Π(X ), dene an operation ◦ p : Π(X ) ×Π(X )

σ ◦p ς = σpς, for∀σ, ς ∈Π(X ).


The unit in (Π( X ); p) is 1◦ p = p− 1. In fact, for θ Π(



( ( )

◦p) p p

∀∈

( p− 1 ◦ p θ = θ ◦ p p− 1 = θ.

For an element σ∈Π(X ), σ− 1◦ p = p− 1σ− 1 p− 1 = ( pσp)− 1. In fa

σ ◦ p ( pσp)− 1 = σpp− 1σ− 1 p− 1 = p− 1 = 1 ◦ p ,

( pσp)− 1 ◦ p σ = p− 1σ− 1 p− 1 pσ = p− 1 = 1 ◦ p .

By denition, we know that (Π( X ); ◦ p) is a group.Notice that if p = 1X , the operation ◦ p is just the compositi

(Π(X ); ◦ p) is the symmetric group Sym(X ) on X . Furthermoropens a general way for constructing multi-groups on permutationus to nd the next result.

Theorem 2.5.2 Let Γ be a permutation group on X , i.e., Γ≺Sm permutations p1, p2, · · ·, pm ∈Γ, (Γ; OP ) with OP = {◦ p, p∈i ≤m}is a permutation multi-group, denoted by G P

X .

Proof First, we check that (Γ; {◦ pi , 1 ≤ i ≤ m}) is an asActually, for ∀σ,ς,τ ∈

G and p, q∈ { p1, p2, · · ·, pm}, we know tha

(τ ◦ p σ) ◦q ς = ( τpσ) ◦q ς = τpσqς

= τp(σ

◦q ς ) = τ

◦ p (σ

◦q ς ).

Similar to the proof of Theorem 2 .5.1, we know that (Γ; ◦ pi ) integer i, 1 ≤ i ≤m. In fact, 1◦ p i

= p− 1i and σ− 1

◦ p i= ( piσpi)− 1 in (G


Denoted by Π (s )(X ) all such s-vectors p(m ) . Let be an



◦a bullet operation of two m-permutations

P (m ) = ( p1, p2, · · ·, pm ),

Q(sm ) = ( q1, q2, · · ·, qm )

on

◦by

P (s) •Q(s ) = ( p1 ◦q1, p2 ◦q2, · · ·, pm ◦qm

Whence, if there are l-operations ◦i , 1 ≤ i ≤ l on X , we obtaitem Π (s) (X ) under these l bullet operations •i , 1 ≤ i ≤ l, dewhere l

1=

{•i|1

≤i

≤l

}.

Theorem 2.5.3 Any s-operation system (H , O) on H witeration ◦i , 1 ≤ i ≤s in O is isomorphic to an s-permutation

Proof For a∈H , dene an s-permutation σa∈Π(s) (H

σa (x) = ( a ◦1 x, a ◦2 x, · · ·, a ◦s x)

for∀x∈H .

Now let π : H →Π(s )(H ) be determined by π(a) = σ

σa (1◦ i ) = ( a

◦1 1◦ i ,

· · ·, a

◦i− 1 1◦ i , a , a

◦i+1 1◦ i ,

· ·we know that for a, b∈H , σa = σb if a = b. Hence, π is a 1

For ∀i, 1 ≤ i ≤s and ∀x∈H , we nd that


According to Theorem 2 .5.3, these algebraic multi-systems



permutation multi-systems, particularly for multi-groups.

Corollary 2.5.1 Any s-group (H , O) with O = {◦i|1 ≤ i ≤s}iss-permutation multi-group (Π(s) (H ); s

1).

Proof It can be shown easily that (Π (s )(H ); s1) is a multi-g

a multi-group.For the special case of s = 1 in Corollary 2 .5.1, we get the w

result on groups.

Corollary 2.5.2(Cayley) A group G is isomorphic to a permutat

As shown in Theorem 2.5.2, many operations can be dened group G, which enables it to be a permutation multi-group, and operations ◦i , 1 ≤ i ≤ s on permutations in Theorem 2 .5.3 ncomposition of permutations. If we choose all ◦i , 1 ≤ i ≤s to be juof permutation, then all bullet operations in s

1 is the same, denoan interesting result following which also implies the Cayley’s resuCorollary 2.5.2.

Theorem 2.5.4 (Π(s) (H ); ) is a group of order (n !)!(n !− s )! .

Proof By denition, we know that

P (s)(x) Q(s ) (x) = ( P 1Q1(x), P 2Q2(x),

· · ·, P s Qs (

for ∀P (s ) , Q (s)∈Π(s )(H ) and ∀x ∈

H . Whence, (1, 1, · · ·, 1) (unit and ( P − (s )) = ( P − 1

1 , P − 12 , · · ·, P − 1

s ) the inverse of P (s ) = (Π(s )(H ); ) Therefore (Π (s) (H ); ) is a group


be a homomorphism



ϕ : (A ; O ) →m

i=1

G P iX i

for sets P 1, P 2, · · ·, P m ≥1 of permutations, i.e., for ∀h∈H

permutation ϕ(h) : x →xh with conditions following hold,

ϕ(h ◦g) = ϕ(h)ϕ(◦)ϕ(g), for h, g∈H i and

Whence, we only need to consider the action of permumulti-sets. Let = ( A ; O ) be a multi-group action on a multiFor a subset Δ

⊂

X , x

∈

Δ, we dene

xG = {xg | ∀g∈G } and G x = {g | xg = x

called the orbit and stabilizer of x under the action of G and

G Δ = {g | xg = x, g∈G for ∀x∈Δ

G (Δ) = {g | Δ g = Δ , g∈G for ∀x∈Δ

respectively. Then we know the result following.

Theorem 2.5.5 Let Γ be a permutation group action on Xmulti-group (Γ; O P ) with P = { p1, p2, · · ·, pm}and pi ∈Γ fThen

(i) |G P X | = |(G P

X )x||xG P X |,∀x∈X ;

(ii ) for ∀Δ ⊂X , ((G P X )Δ , OP ) is a permutation mul


In fact, for

∀

h

∈

Γ, let xh = xk , 1 k m. Then xh = xgk , i.e., x



∀∈ ≤ ≤we get that hg− 1k ∈Γx , namely, h∈gkΓx .

For integers i,j , i = j , there are must be giΓx ∩g j Γx = ∅.exist h1, h2∈Γx such that gih1 = g j h2. We get that xi = xgi = xa contradiction.

Therefore, we nd that

|G P X | = |Γ| = |Γx ||xΓ| = |(G P

X )x ||xG P X |.

This is the assertion ( i). For (ii ), notice that ( G P X )Δ = Γ Δ

permutation group. Applying Theorem 2 .5.2, we nd it.Particularly, for a permutation group Γ action on Ω, i.e., all

i ≤m, we get a consequence of Theorem 2.5.5.Corollary 2.5.3 Let Γ be a permutation group action on Ω. The

(i) |Γ| = |Γx ||xΓ|,∀x∈Ω;

(ii ) for ∀Δ ⊂Ω, ΓΔ is a permutation group.

Theorem 2.5.6 Let Γ be a permutation group action on X and multi-group (Γ; O P ) with P = { p1, p2, · · ·, pm}, pi ∈Γ for integeOrb(X ) the orbital sets of G P

X action on X . Then

|Orb(X )| =1

|G P X | p∈G P

X

|Φ( p)|,

where Φ( p) is the xed set of p, i.e., Φ( p) = {x∈X |x p = x}.

Proof Consider a set E = {( p,x) ∈G P

X ×X |x p = x}. Th


that

|(G P

X)y

|=

|(G P

X)x

i |for any element in x

G P X

i, 1 i

|Or



| X | | X i | i ≤ ≤ |to obtain that

p∈G P X

|Φ( p)| =x∈X

(G P X )x =

|Orb (X ) |

i=1y∈x

G P X

i

|(

=|Orb (X ) |

i=1|x

G P X

i ||(G P X )x i | =

|Orb (

i=1

= |Orb(X )||G P X |

by applying Theorem 2 .5.5. This completes the proof.

For a permutation group Γ action on Ω, i.e., all pi = 1X

the famous Burnside’s Lemma by Theorem 2 .5.6.

Corollary 2.5.4(Burnside’s Lemma) Let Γ be a permutatiThen

|Orb(Ω)| = 1|Γ| g∈Γ |Φ(g)|.

A permutation multi-group G P X is transitive on X if |O

elements x, y ∈X , there is an element g∈G P

X such that xknow formulae following by Theorems 2.5.5 and 2.5.6.

|G P X | = |X ||(G P

X )x | and |G P X | =

p∈G P X

|Φ


(ii ) G P X is k-transitive on X if and only if (G P

X )x is (k 1)-tra



−Proof If Γ is k-transitive on X , it is obvious that Γ is ( kX itself. Conversely, if Γx is (k −1)-transitive on X \ {x}, then(x1, x2, · · ·, xk) and (y1, y2, · · ·, yk), there are elements g1, g2 ∈Γthat

xg11 = x, yg2

1 = x and (xg1i )h = yg2

i

for any integer i, 2 ≤ i ≤k. Therefore,

xg1 hg − 12

i = yi , 1 ≤ i ≤k.

We know that Γ is ‘ k-transitive on X . This is the assertion oBy denition, G P

X is k-transitive on X if and only if Γ isk-trais (k −1)-transitive on X \ {x}by (i), which is the assertion of (

Applying Theorems 2 .5.5 and 2.5.7 repeatedly, we get an quence for k-transitive multi-groups.

Theorem 2.5.8 Let G P X be a k-transitive multi-group and Δ ⊂

Then

|G P X | = |X |(|X | −1) · · ·(|X | −k + 1 |(G P

X )Δ |.

Particularly, a k-transitive multi-group G P X with |G P X | = |X |(k + 1 | is called a sharply k-transitive multi-group . For example, chwith |X | = n, i.e., the symmetric group S n and permutations pi


G P X is primitive if there are no G P

X -admissible relations R onR {( )| ∈X} i th t i i ll l ti Th t



R = {(x, x )|x∈X }, i.e., the trivially relations. The next resuof primitive multi-groups.

Theorem 2.5.9 Every k-transitive multi-group G P X is primi

Proof Otherwise, there is a G P X -admissible relations R

X ×X and R = {(x, x )|x∈X }. Whence, there must existand x = y. By assumption, G P

X is k-transitive on X , k ≥ 2an element g∈

G P X such that xg = x and yg = z. This fact

for ∀z ∈X by denition. Notice that R is an equivalen

∀z1, z2∈X , we get (z1, x), (x, z2)∈R. Thereafter, ( z1, z2)∈

a contradiction.There is a simple criterion for determining which perm

primitive by maximal stabilizers following.

Theorem 2.5.10 A transitive multi-group G P X is primitive i

element x∈X such that p∈(G P X )x for ∀ p∈P and if there

group (H; O P ) enabling ((G P X )x ; O P )≺(H; O P )≺G P X , then

or G P X .

Proof If (H; O P ) be a multi-group with (( G P X )x ; O P )≺

element x∈X , dene a relation

R =

{(xg, xg◦ h )

|g

∈

G P

X, h

∈H.

for a chosen operation ◦ ∈O P . Then R is a G P X -admissible

fact, it is G P X -admissible, reexive and symmetric by denit


∀

g

∈

G P X and h

∈H. Particularly, for g = 1 ◦ , we nd that xh =

(H; O ) = (( G P ) ; O )



∀∈ ∈H(H; O P ) = (( G P X )x ; O P ).

If R = X ×X , then (x, x f )∈R for ∀f ∈G P

X . In this caseg∈

G P X and h∈Hsuch that x = xg, xf = xg◦ h . Whence, g∈((G P

X

and g− 1 ◦ h− 1 ◦ f ∈((G P X )x ; O P ) ≺(H; O P ). Therefore, f ∈

((G P X ); O P ).

Conversely, assume R to be a G P X -admissible equivalent rel

an element x∈X such that p∈(G P X )x for∀ p∈P , ((G P

X )x ; O P )implies that ( H; O P ) = (( G P

X )x ; O P ) or ((G P X ); O P ). Dene

H= {h∈G P

X | (x, x h )∈R }.

Then (H; O P ) is multi-subgroup of G P X which contains a multi-subg

by denition. Whence, ( H; O P ) = (( G P X )x ; O P ) or G P

X .

If (H; O P ) = ( ( G P X )x ; O P ), then x is only equivalent to it

transitive on X , we know that R = {(x, x )|x∈X }.

If (H; O P ) = G P X , by the transitiveness of G P

X on X again,

X ×X . Combining these discussions, we conclude thatG P

X is priChoose p = 1 X for each p∈P in Theorem 2.5.10, we get a

in classical permutation groups following.

Corollary 2.5.5 A transitive group Γ is primitive if and only if tx∈X such that a subgroup H with Γx≺H ≺Γ hold implies tha

Now let Γ be a permutation group action on a set X and P shown in Theorem 2.5.2 that (Γ; O P ) is a multi-group if P ⊂Γ. Tsay if not all p∈P are in Γ? For this case we introduce a new mu


(i) ΓP X = Γ

∪

P Γ , particularly, ΓP X = G P

X if and

(ii ) f b Λ f Γ h i b P



∪(ii ) for any subgroup Λ of Γ, there exists a subset P ⊂

ΛP X ; O P = ΓP

X .

Proof By denition, for ∀a, b∈Γ and p∈P , we know

a ◦ p b = apb.

Choosing a = b = 1 Γ , we nd that

a ◦ p b = p,

i.e., Γ

⊂

Γ. Whence,

Γ∪P Γ⊂ΓP X .

Now for∀gi∈Γ and p j ∈P , 1 ≤ i ≤ l + 1, 1 ≤ j ≤ l, w

g1 ◦ p1 g2 ◦ p2 · · · ◦ pl gl+1 = g1 p1g2 p2 · · · plg

which means that

ΓP X ⊂Γ∪P Γ .

Therefore,

ΓP X = Γ∪P Γ .

Now if ΓP X = G

P X , i.e., Γ∪P Γ = Γ, there must be

Theorem 2 .5.2, this concludes the assertion ( i).For the assertion ( ii ), notice that if P = Γ \ Λ, we get


set X . If Γ is k-transitive on X , choose P =

∅

enabling the conclwise assume these orbits of Γ action on X to be O1 O2 Os wh



∅wise, assume these orbits of Γ action on X to be O1, O2, · · ·, Os , whConstruct a permutation p∈Π(X ) by

p = ( x1, x2, · · ·, xs ),

where xi ∈Oi , 1 ≤ i ≤ s and let P = { p} ⊂Sym(X ). Applyin

we know that ΓP X = Γ∪P Γ is transitive on X with |P | = 1.Now for an integer k, if ΓP 1

X is k-transitive with |P 1| ≤k, lethese orbits of the stabilizer ΓP 1

X y1 y2 ···ykaction on X \ {y1, y2, ·

a permutation

q = ( z1, z2, · · ·, zl),

where zi ∈Oi , 1 ≤ i ≤ l and let P 2 = P 1∪ {q}. Applying Theowe nd that ΓP 2

X y1 y2 ···ykis transitive on X \ {y1, y2, · · ·, yk}, whe

Therefore, ΓP 2X is (k + 1)-transitive on X with |P 2| ≤k + 1 by T

Applying the induction principle, we get the conclusion.

Notice that any k-transitive multi-group is primitive if k

≥2

We have a corollary following by Theorem 2 .5.12.

Corollary 2.5.6 Let Γ be a permutation group action on a set P ∈Π(X ) such that ΓP

X is primitive.

§2.6 COMBINATORIAL ALGEBRAIC SYSTEMS

2 6 1 Algebraic Multi-System An algebraic multi-system is

Sec.2.6 Combinatorial Algebraic Systems

for k = 1 , 2 are homomorphic if there is a mapping o : A

is a homomorphism for any integer i, 1 ≤ i ≤ m. By this d



is a homomorphism for any integer i, 1 ≤ i ≤ m. By this dexistent conditions for homomorphisms on algebraic multi-sy

Theorem 2.6.1 There exists a homomorphism from an (A 1; O 1) to (A 2; O 2), where A k =

m

i=1H k

i and O k =m

i=1 Oki f

if there are homomorphisms η1, η2, · · ·, ηm on (H 1

1 ;O11), (H

such that

ηi |H 1i ∩H 1

j= η j |H 1

i ∩H 1j

for any integer 1

≤i, j

≤m.

Proof By denition, if there is a homomorphism o : (A

opi is a homomorphism on ( H 1i ;O1

i ) for any integer i, 1 ≤ i

On the other hand, if there are homomorphisms η1, η(H 1

2 ;O12), · · ·, (H 1

m ;O1m ), dene a mapping o : (A 1; O 1) →(

if a

∈

H 1i . Then it can be checked immediately that o is a

Let o : (A 1; O 1) →(A 2; O 2) be a homomorphism with aation ◦ ∈O 2. Similar to the case of multi-operation systemkernel Ker(o) by

Ker(o) =

{a

∈

A 1 |

o(a) = 1◦

for

∀◦∈Then we have the homomorphism theorem on algebraic m




The advantage of this diagram on systems is that we caedge in GL [A], if its vertices are a,c with a label ◦b and vice



g ◦example, the labeled graph GL [Z 4] of an Abelian group Z 4 i

¹

¹

¹ ¹

Á

Ê

«

0 1

23

+0

+0

+1

+1

+1

+1

+3

+3

+3

+3 +2

+2

+2

+2

Fig. 2.6.2

Some structure properties on these diagrams GL [A] of sfollowing.

Property 1. The labeled graph GL

[A] is connected if anpartition A = A1 A2 such that for ∀a1∈A1, ∀a2∈A2, thea1 ◦a2 in (A;◦).

If GL [A] is disconnected, we choose one component CDene A2 = V (GL [A]) \ V (C ). Then we get a partition

∀a1∈A1,∀a

2∈A2, there are no denition for a1 ◦a2 in (A

Property 2. If there is a unit 1A in (A;◦), then there exish th t th l b l th dg (1 ) i


Property 5. If the cancelation law holds in (A;

◦), i.e., for

∀

a,b,cthen b = c, then there are no parallel multiple 2-edges in GL [A].



∀

These properties 2 −5 are gotten by denition. Each of these(1), (2), (3) and (4) in Fig.2 .6.3.

a b

1A

(1)

◦a ◦b

a

1A

◦a ◦a− 1

a b

◦b ◦a

a

◦b

(2) (3) (4

Fig. 2.6.3

Now we consider the diagrams of algebraic multi-systems. Lalgebraic multi-system with

A =m

i=1H i and O =

m

i=1 Oi

such that ( H i ;Oi) is a multi-operation system for any integer i,the operation set Oi = {◦ij |1 ≤ j ≤n i}. Dene a labeled graphwith ( A ; O ) by

GL [A ] =m

i=1

n i

j =1GL [(H

i;

◦ij)],

where GL [(H i ;◦ij )] is the associated labeled graph of ( H i ;◦ij )1 ≤ j ≤n ij The importance of GL [A ] is displayed in the next re


A 2 with ω : O 1

→O 2 such that for

∀

a, b

∈

A 1 and

◦1

∈

O 1, th

◦2∈O 2 with the equality following hold,



∀ ∈ ∈∈

ω(a ◦1 b) = ω(a) ◦2 ω(b).

Not loss of generality, assume a ◦1 b = c in (A 1; O 1). Then flabel

◦1b in GL [A 1], there is an edge (ω(a), ω(c)) with a labe

ω is an equivalence from GL [A 1] to GL [A 2]. Therefore, GL [A

Conversely, if GL [A 1] ∼= GL [A 2], let be a such equiGL [A 2], then for an edge (a, c) with a label ◦1b in GL [A 1]that ( ω(a), ω(c)) with a label ω(◦1)ω(b) is an edge in GL [A 2

ω(a ◦1 b) = ω(a)ω(◦1)ω(b),

i.e., ω : A 1 →A 2 is an isomorphism from ( A 1; O 1) to (A 2; O

Generally, let ( A 1; O 1), (A 2; O 2) be two algebraic multi-slabeled graphs GL [A 1], GL [A 2]. A homomorphism ι : GL [A

ping ι : V (GL [A 1])

→V (GL [A 2]) and ι : O 1

→O 2 such th

with a label ι(◦)ι(b) for ∀(a, c) ∈E (GL [A 1]) with a labelhomomorphisms of labeled graphs following.

Theorem 2.6.4 Let (A 1; O 1), (A 2; O 2) be algebraic multm

i=1H k

i , O k =m

i=1 Oki for k = 1 , 2 and ι : (A 1; O 1) →(A 2

Then there is a homomorphism ι : GL [A 1] →GL [A 2] from Gby ι.

Proof By denition we know that o : V(GL [A 1])


E (Γ) =

{(H i , H j )

|∃a

∈

H i , b

∈

H j with ( a, b)

∈

E (GL [A ]) fo

We obtain conditions for an algebraic multi-system with a gra



{ | ∈ ∈ ∈We obtain conditions for an algebraic multi system with a grain the following.

Theorem 2.6.5 Let (A ; O ) be an algebraic multi-system. Then i

(i) a circuit multi-system if and only if there is arrangemen

for H 1, H 2, · · ·, H m such that

H li − 1H li = ∅,

H liH li +1 = ∅

for any integer i(mod m), 1 ≤ i ≤m but

H liH lj = ∅

for integers j = i −1, i , i + 1( mod m);(ii ) a star multi-system if and only if there is arrangement

H 1, H 2, · · ·, H m such that

H l1H li = ∅but H li

H lj = ∅

for integers 1 < i, j ≤m, i = j .(iii ) a tree multi-system if and only if any subset of A is n

system under operations in O .

Proof By denition, these conditions really ensure a circumulti-system, and conversely, a circuit, star, or a tree multi-system

diti ti l


Summarizing previous discussion, we can sketch the diaas follows.



Theorem 2.6.6 Let (A ; O ) be a multi-group with A =m

i=

{◦ij , 1 ≤ j ≤ n i}and (H i ;◦ij ) a group for integers i,j, 1Then its diagram GL [A ] is

GL [A ] =m

i=1

n i

j =1

K [H i ;◦ij ].

Corollary 2.6.1 The diagram of a eld (H ; + , ◦) is a union

attached with 2 loops at each vertex.Corollary 2.6.2 Let (A ; O ) be a multi-group. Then GL [A

only if C Γ is hamiltonian.

Proof Notice that C Γ is an resultant graph in GL [A ] shri

to a vertex H i for 1

≤i

≤m by denition. Whence, C Γ is h

hamiltonian.

Conversely, if C Γ is hamiltonian, we can easily nd a GL [A ] by applying Theorem 2 .6.6.

2.6.3 Cayley Diagram. Besides these diagrams of mul

Theorem 2 .4.5, these is another diagram for a multi-systemcalled Cayley diagrams of multi-systems dened in the follow

A multi-system ( A ; O ) is nitely generated if there are n


For the case of multi-groups ( A ; O ), some elementary properin [Mao3], particularly, if ( A ; O ) is a group, these Cayley diagram



the Cayley graphs of nite groups introduced in graph theory, whichbehaviors. For multi-groups, the following structure result is obtain

Theorem 2.6.7 Let Cay(Γ : S ) be a Cayley diagram of a multi-

Γ =m

i=1Γ

i, O =

{◦i|1

≤i

≤m

}and S =

m

i=1S

i, Γ = S

i;

◦ifor 1

Cay(Γ : S ) =n

i=1Cay(Γ i : S i).

As we known, few results can be found for Cayley diagrams

on publications unless [Mao3]. So, to nd out such behaviors for good topic for researchers.

§2.7 REMARKS

2.7.1 The original form of the combinatorial conjecture for mamathematical science can be reconstructed from or made by combabbreviated to CCM Conjecture in [Mao4] and [Mao10]. Its impbinatorial notion for entirely developing mathematical sciences, wenormous creative power for modern mathematics and physics.

2.7.2 The relation of Smarandache’s notion with LAO ZHI ’s pointed out by the author in [Mao19], reported at the 4th Internaon Number Theory and Smarandache Problems of Northwest of Ch

Sec.2.7 Remarks

works in [Mao7] to multi-modules. There are many trends or tshould be researched, such as extending those of results in

l



spaces to multi-systems.

2.7.4 Considering the action of multi-systems on multi-sets lem, which requires us to generalize permutation groups groups. This kind of action, i.e., multi-groups on nite min [Mao20]. The construction in Theorems 2 .5.1 and 2.5.2abstract multi-groups. But in fact, an action of a multi-group dependent on their combinatorial structures. This means geaction of multi-groups must consider their underlying labelcandidate topic for postgraduate students.

2.7.5 The topic discussed in Section 2 .6 can be seen as binatorial notion to classical algebra. In fact, there are macombinatorial algebraic systems , in algebra or combinatorics.underlying combinatorial structure G, what can we say abouSimilarly, what can we know on its graphical structure, such has a hamiltonian circuit, or a 1-factor ? When it is regular ?Cayley diagrams Cay(A : S ) of multi-systems ( A ; O ), parwhat can we know on their structure ? Determine those properCay(A : S ) which Cayley graphs of nite groups have.



CHAPTER 3.

Smarandache manifolds

A Smarandache geometry is a geometrical Smarandache system, wmeans that there is a Smarandachely denied axiom in this geomcal system, i.e., both validated and invalidated, or just invalidatedin multiple distinct ways, which is a generalization of classical gtries. For example, these Euclid, Lobachevshy-Bolyai-Gauss anmannian geometries maybe united altogether in a same space by

Smarandache geometries. A Smarandache geometry can be eithetially Euclidean and partially non-Euclidean, or non-Euclidean connewith the relativity theory because they include Riemannian geoma subspace, also with the parallel universes in physics because thebine separate spaces into one space too. A Smarandache manifol

topological or differential manifold which supports a Smarandachometry. For an introduction on Smarandache manifolds, Sectionand 3.2 present the fundamental of algebraic topology and differ

Sec.3.1 Topological Spaces

§3.1 TOPOLOGICAL SPACES

3.1.1 Topological Space. A topological space is a set S to



p g p p g pC of subsets called open sets such that

(T1) ∅∈C and S ∈

C ;

(T2) if U 1, U 2∈C , then U 1 ∩U 2∈

C ;

(T3) the union of any collection of open sets is open.

Example 3.1.1 Let R be the set of real numbers. We hintervals ( a, b) for a ≤b,a,b∈R in elementary mathematicsto be a union of nite open intervals. Then it can be shownhold. Consequently, R is a topological space.

A set V is closed in a topological space S if S \ V is opa topological space S , the relative topological on A in S is d

C A = {U A | ∀U ∈C }.

Applied these identities

(i) ∅∩A = ∅, S ∩A = A;(ii ) (U 1 ∩U 2) ∩A = ( U 1 ∩A) ∩(U 2 ∩A);(iii )

α(U α A) = (

αU α ) A

in Boolean algebra, we know that C A is indeed a topology subspace with topology C

Aof S .

For a point u in a topological space S , its an open neighset U such that u∈U and a neighborhood in S is a set cont

90 Chap.3

such that for every neighborhood U of u, there is an integer N implies xn ∈U , then we say that {un}converges to u or u is a lim

Let S and T be topological spaces with ϕ : S T a mappin



Let S and T be topological spaces with ϕ : S →T a mappinat u∈S if for every neighborhood V of ϕ(u), there is a neighborthat ϕ(U )⊂V . Furthermore, if ϕ is continuous at any point ucontinuous mapping .

Theorem 3.1.1 Let R, S and T be topological spaces. If f : R →are continuous at x∈R and f (x)∈S , then the composition mapis also continuous at x.

Proof Since f and g are respective continuous at x ∈R aany open neighborhood W of point g(f (x))

∈

T , g− 1(W ) is openef (x) in S . Whence, f − 1(g− 1(W )) is an opened neighborhood of xTherefore, g(f ) is continuous at x.

The following result, usually called Gluing Lemma , is very usecontinuous mappings on a union of spaces.

Theorem 3.1.2 Assume that a space X is a nite union of clon

i=1X i . If for some space Y , there are continuous maps f i : X i

overlaps, i.e., f i |X i X j = f j |X i X j for all i, j , then there exists a uf : X →Y with f |X i = f i for all i.

Proof Obviously, the mapping f dened by

f (x) = f i(x), x∈X i


By assumption, each f i is continuous. We know thatWhence, f − 1(U ) is open in X , i.e., f is continuous on X .

A collection C⊂2X is called a cover of X if



A collection C ⊂2 is called a cover of X if

C ∈CC = X.

If each set in C is open, C is called an opened cover and if | nite cover of X . A topological space is compact if there existopened cover and locally compact if it is Hausdorff with a comits each point. As a consequence of Theorem 3 .1.2, we can to ascertain continuous mappings shown in the next.

Corollary 3.1.1 Let {A1, A2, · · ·, An}be a nite opened c

X →Y is continuous constrained on each Ai , 1 ≤ i ≤ n, mapping.

Two topological spaces S and T are homeomorphic if thmapping ϕ : S →T such that its inverse ϕ

− 1 : T →S ismapping ϕ is called a homeomorphic or topological mappin

logical spaces is said topological invariant if it is not variablmappings. In topology, a fundamental problem is to classifequivalently, to determine wether two given spaces are homeohave known many homeomorphic spaces, particularly, spaces example.

Example 3.1.2 Each of the following topological space pai

(1) a Euclidean space R n and an opened unit n-ball B n

2 2 1

92 Chap.3

for

∀

(x1, x2,

· · ·, xn )

∈

B n with an inverse

f − 1(x1 x2 xn ) =(x1, x2, · · ·, xn )



f (x1, x2, · · ·, xn ) 1 + x2

1 + x22 + · · ·+ x2

n

for∀(x1, x2, · · ·, xn )∈R n .

For the case (2), let ( x0, y0, z0) be the north pole with coord

the Euclidean plane R 2 be a plane containing the circle {(x, y) | xa homeomorphic mapping g from S 2 to R 2 is dened by

g(x,y,z ) = (x

1 −z,

y1 −z

).

The readers are required to nd a homeomorphic mapping in

3.1.2 Metric Space. A metric space (M ; ρ) is a set M associfunction ρ : M ×M →R+ = {x | x∈R, x ≥0}with conditions fofor∀x,y,z ∈M .

(1)( deniteness ) ρ(x, y) = 0 if and only if x = y;

(ii )(symmetry ) ρ(x, y) = ρ(y, x );(iii )( triangle inequality ) ρ(x, y) + ρ(y, z ) ≥ρ(x, z ).

For example, the standard metric function on a Euclidean spaby

ρ(x , y) = n

i=1

(xi −yi)

for∀x = ( x1 x2 x ) and y = ( y1 y2 y )∈R n


D 1 (x1)

∩D 2 (x2) since for

∀

y

∈

D x (x),

ρ(y, x 1) ≤ρ(y, x ) + ρ(x, x 1) < x + ρ(x, x 1)



Similarly, we know that ρ(y, x ) < 2. Therefore, D x (xwe nd that

D 1 (x1) ∩D 2 (x2) = x∈D 1 (x1 )∩D 2 (x2 ) D x

i.e., it enables the condition (T3) hold.Let (M ; ρ) be a metric space. For a point x∈M and A

inf {d(x, a )|a∈A}if A = ∅, otherwise, ρ(x,∅) = ∞. The diis dened by diam( A) = sup {ρ(x, y)|x, y∈A}. Now let x1, x

sequence in a metric space (M ; ρ). If there is a point x∈M sthere is an integer N implies that ρ(xn , x) < providing nsequence {xn}converges to x or x is a limit point of {xn}, dThe following result, called Lebesgue lemma , is a useful resu

Theorem 3.1.3(Lebesgue Lemma) Let

{V α

|α

∈

Π

}be an op

metric space (M ; ρ). Then there exists a positive number λof diameter less than λ is contained in one of member of {Vλ is called the Lebesgue number.

Proof The proof is by contradiction. If there no suchchoosing numbers 1, 2,

· · ·with lim

n →∞n = 0, we con cons

A2⊃ · · ·with diameter diam( An ) = n , but each An is not ain {V α |α∈Π}for n ≥ 1. Whence, lim

n →∞diam( An ) = 0. Cho

A d ∈ A Th li

94 Chap.3

3.1.3 Fundamental Group. A topological space S is conn

no open subspaces A and B such that S = A∪B with A, B =for characterizing connectedness is by arcwise connectedness. Cert



g yspaces are arcwise connected in most cases considered in topology

Denition 3.1.1 Let S be a topological space and I = [0, 1]⊂is a continuous mapping a : I

→S with initial point a(0) and en

S is called arcwise connected if every two points in S can be joinAn arc a : I →S is a loop based at p if a(0) = a(1) = p∈S . Ae : I →x∈S , i.e., mapping each element in I to a point x, usuloop.

For example, let G be a planar 2-connected graph on R 2 andspace consisting of points on each e∈E (G). Then S is a arcwisby denition. For a circuit C in G, we choose any point p on C .e p in S based at p.

Denition 3.1.2 Let a and b be two arcs in a topological space SA product mapping a

·b of a with b is dened by

a ·b(t) =a(2t), if 0≤ t ≤ 1

2 ,b(2t −1), if 1

2 ≤ t ≤1

and an inverse mapping a = a(1 −t) by a.

Notice that a

·b : I

→S and a : I

→S are continuous b

Whence, they are indeed arcs by denition, called the product arthe inverse arc of a. Sometimes it is needed to distinguish the orie


H : I ×I →S

such that H (t, 0) = a(t), H (t, 1) = b(t) for ∀t∈I , then a an



such that H (t, 0) a(t), H (t, 1) b(t) for ∀t∈I , then a andenoted by a b and H a homotopic mapping from a to b.

Theorem 3.1.4 The homotopic is an equivalent relation,to an arc a is an equivalent arc class, denoted by [a].

Proof Let a,b,c be arcs in a topological space S , a b anmappings H 1 and H 2. Then

(i) a a if choose H : I ×I →S by H (t, s ) = a(t) fo

(ii ) b a if choose H (t, s ) = H 1(t, 1 −s) for ∀s, t

continuous;(iii ) a c if choose H (t, s ) = H 2(H 1(t, s )) for ∀s, t ∈

3.1.1 for the continuity of composition mappings.

Theorem 3.1.5 Let a,b,c and d be arcs in a topological spa

(i) a b if a b;(ii ) a ·b c ·d if a b, c d with a ·c an arc.

proof Let H 1 be a homotopic mapping from a to b. DenH : I × I →S by H (t, s ) = H 1(1 − t, s ) for ∀t, s ∈H (t, 0) = a(t) and H (t, 1) = b(t). Whence, we get that a

(i).For (ii ), let H 2 be a homotopic mapping from c to d.

I ×I →S by

96 Chap.3

Then we know that π1(S, x 0) is a group shown in the next.

Theorem 3.1.6 π1(S, x 0) is a group.



Proof We check each condition of a group for π1(S, x 0). First,the operation ◦ since [a]◦ [b] = [a ·b] is an equivalent class of loofor∀[a], [b]∈π1(S, x 0).

Now let a,b,c : I →S be three loops based at x0. By Denitthat

(a ·b) ·c(t) =⎧⎪⎪⎨

⎪⎪⎩

a(4t), if 0 ≤ t ≤ 14 ,

b(4t −1), if 14 ≤ t ≤ 1

2 ,c(2t −1), if 1

2 ≤ t ≤1.

and

a ·(b ·c)(t) =⎧⎪⎪⎨⎪

⎪⎩

a(2t), if 0≤ t ≤ 12 ,

b(4t −2), if 12 ≤ t ≤ 3

4 ,c(4t −3), if 3

4 ≤ t ≤1.

Consider a function H : I ×I →S dened by

H (t, s ) =⎧⎪⎪⎨⎪⎪

⎩

a( 4t1+ s ), if 0≤ t ≤ s+1

4 ,b(4t −1 −s), if s+1

4 ≤ t ≤ s+24 ,

c(1 − 4(1− t )2− s ), if s+2

4 ≤ t ≤1.

Then H is continuous by applying Corollary 3 .1.1, H (t, 0) = (H (t, 1) = ( a ·(b·c))( t). Consequently, we know that ([ a]◦[b]) ◦[

Now let ex0 : I →x0 ∈S be the point loop at x0. Then it


Let S be a topological space, x0, x1 ∈S and £ an a

∀[a] ∈π1(S, x 0), we know that £ ◦ [a] ◦ £ − 1∈π1(S, x 1)

Whence, the mapping £ # = £ ◦ [a]◦£ − 1 : π1(S, x 0) →π1(S



¶

x0

x1£

[a]

Fig. 3.1.1

Theorem 3.1.7 Let S be a topological space. If x0, x1∈S ato x1 in S , then π1(S, x 0) ∼= π1(S, x 1).

Proof We have known that £ # : π1(S, x 0) →π1(S,π1(S, x 0), [a] = [b], we nd that

£ # ([a]) = £ ◦[a]◦£ − 1 = £ ◦ [b]◦£ − 1 = £

i.e., £ # is a 1

−1 mapping. Let [c]

∈

π1(S, x 0). Then

£ # ([a]) ◦£ # ([c]) = £ ◦ [a]◦£ − 1 ◦£ ◦ [b]◦£ − 1 = £ ◦= £ ◦ [a]◦ [b]◦£ − 1 = £ # ([a]◦ [b]).

Therefore, £ # is a homomorphism.Similarly, £ − 1

# = £ − 1◦[a]◦£ is also a homomorphism froand £ − 1

# ◦£ # = [ex1 ], £ # ◦£ − 1# = [ex0 ] are the identity mapp

98 Chap.3

π1(S n ) =ex0 , if n ≥2,Z, if n = 2 ,



Z, if n 2 ,

seeing [Amr1] or [Mas1] for details.

Theorem 3.1.8 Let G be an embedded graph on a topological

spanning tree in G. Then π1(G) = T + e | e∈E (G) \ {e} .Proof We prove this assertion by induction on the number o

n = 0, G is a bouquet, then each edge e is a loop itself. A closcombination of edges e in E (G), i.e., π1(G) = e | e∈E (G) in

Assume the assertion is true for n = k, i.e., π1(G) = T + eConsider the case of n = k + 1. For any edge e∈E (T ), we consigraph G/ e, which means continuously to contract e to a point v inon G passes or not through e in G is homotopic to a walk passes oG/

e for κ(T ) = 1. Therefore, we conclude that π1(G) = T + e

by the induction assumption.

For calculating fundamental groups of topological spaces, the and Van-Kampen theorem is useful.

Theorem 3.1.9(Seifert and Van-Kampen) Let S 1, S 2 be two open space S with S = S 1∪S 2. If there S, S 1, S 2 and S 1 ∩S 2 are nconnected, then for ∀x0∈S ,

π1(S, x 0) ∼=π1(S 1, x0)π1(S 2, x0)

(i1)π ([a])(i2)π [a− 1] | [a]∈π1(S 0, x0)



100 Chap.3

for any real number t ∈R . Then the pair ( R , p) is a covering

circle S 1. In this example, each opened subinterval on S 1 servesneighborhood.



Denition 3.1.5 Let S, T be topological spaces, x0 ∈S, y0 ∈T (S, x 0) a continuous mapping. If (S, p) is a covering space of S , xand there exists a mapping f l : (T, y0)

→(S, x0) such that

f = f l ◦ p,

then f l is a lifting of f , particularly, if f is an arc, f l is called a

Theorem 3.1.10 Let (S, p) be a covering space of S , x0 ∈XThen there exists a unique lifting arc f l : I

→S with initial poi

f : I →S with initial point x0.

Proof If the arc f were contained in an arcwise connected let V be an arcwise connected component of p− 1(U ) which contawould exist a unique f l in V since p topologically maps V onto U

Now let

{U i

}be a covering of S by elementary neighborhoods

is an opened cover of the unit interval I , a compact metric space. n so large that 1 /n is less than the Lebesgue number of this covinterval I into these closed subintervals [0 , 1/n ], [1/n, 2/n ], · · ·, [(n

According to Theorem 3 .1.3, f maps each subinterval into an eborhood in

{U i

}. Dene f l a successive lifting over these subinte

edness is conrmed by Corollary 3.1.1.

For the uniqueness, assume f l1 and f l2 be two liftings of an a


Similarly, if A is closed, a contradiction can be also nd

closed and opened. Since A = ∅, we nd that A = I , i.e., f l1

Theorem 3.1.11 Let (S, p) be a covering space of S , x0∈S



( p) g p ∈(i) the induced homomorphism p∗ : π(S, x0) →π(S, x 0

(ii ) for x∈ p− 1(x0), the subgroups p∗π(S, x0) are exac

subgroups of π(S, x 0).Proof Applying Theorem 3 .10, for x0 ∈S and p(x0)

mapping on loops from S with base point x0 to S with bLi : I →S, i = 1 , 2 be two arcs with the same initial pointif pL1 pL2, then L1 L2.

Notice that pL1 pL2 implies the existence of a conI ×I →S such that H (s, 0) = pl1(s) and H (s, 1) = pL2(sof Theorem 3.10, we can nd numbers 0 = s0 < s 1 < · · ·<t1 < · · ·< t n = 1 such that each rectangle [ s i− 1, s i] ×[t j −

elementary neighborhood in S by H .

Now we construct a mappingG

:I ×I →S

withpGby the following procedure.

First, we can choose G to be a lifting of H over [0, s1]×rectangle into an elementary neighborhood of p(x0). Then wof G successively over the rectangles [s i− 1, s i] ×[0, t1] for icare that it is agree on the common edge of two successive reus to get G over the strip I ×[0, t1]. Similarly, we can extendI ×[t1, t2], [t2, t3], · · ·, etc.. Consequently, we get a lifting H

102 Chap.3

Theorem 3.1.12 If (S, p) is a covering space of S , then the set

same cardinal number for all x∈S .

Proof For any points x1 and x2∈S , choosing an arc f in S w



and terminal point x2. Applying f , we can dene a mapping Ψ :by the following procedure.

For ∀y1 ∈p− 1(x1), we lift f to an arc f l in S with initial p

pf l = f . Denoted by y2 the terminal point of f l. Dene Ψ(y1) =By applying the inverse arc f − 1, we can dene Ψ− 1(y2) = y

way. Therefore, ψ is a 1−1 mapping form p− 1(x1) to p− 1(x2).The common cardinal number of the sets p− 1(x) for x∈S is

of sheets of the covering space (S, p) on S . If | p− 1(x)| = n for x

is an n-sheeted covering.We present an example for constructing covering spaces of g

assignment.

Example 3.1.3 Let G be a connected graph and (Γ; ◦) a groue∈E (G), e = uv, an orientation on e is an orientation on e from

e = ( u, v) , called plus orientation and its minus orientation , froby e− 1 = ( v, u ). For a given graph G with plus and minus orientaa voltage assignment on G is a mapping α from the plus-edges Γ satisfying α(e− 1) = α− 1(e), e∈E (G). These elements α(e), evoltages, and ( G, α ) a voltage graph over the group (Γ; ◦).

For a voltage graph ( G, α ), its lifting Gα

= ( V (Gα

), E (Gα

)by


in Fig.3.1.2.

uu0

u1



w

10

0(G, α )

vv0

w0

w1

Gα

Fig. 3.1.2

We can nd easily that there is a unique lifting path in Γx for each path with an initial point x in Γ, and for

∀

x

∈

Γ,

Let (S 1, p1) and (S 2, p2) be two covering spaces of S . Wif there is a continuous mapping ϕ : (S 1, p1) →(S 2, p2) suticularly, if ϕ : (S, p) →(S, p), we say ϕ an automorphism oonto itself. If so, according to Theorem 3 .1.11, p1∗π(S 1, x1

are conjugacy classes in π(S, x 0). Furthermore, we know the

Theorem 3.1.13 Two covering spaces (S 1, p1) and (S 2, p2

and only if for any two points x1 ∈S 1, x2 ∈S 2 with p1(x1

subgroups p1∗π(S 1, x1) and p2∗π(S 1, x2) belong to a same con

3.1.5 Simplicial Homology Group. A n-simplex sEuclidean space is a set

n +1 n +1

104 Chap.3

Usually, its underlying space is dened by |K | =s∈K

s, i.e., th

simplexes of K . See Fig.3.1.3 for examples. In other words, an ua multi-simplex . The maximum dimensional number of simplex idi i l f K d t d b di K



dimensional of K , denoted by dim K .

simplicial complex non-simplic

Fig. 3.1.3

A topological space P is a polyhedron if there exists a simpand a homomorphism h : |K | →P . An orientation on a simplia partial order on its vertices whose restriction on the vertices of a

is a linear order. Notice that two orientations on a simplex are vertex permutations are different on an even permutation. Whencetwo orientations on a simplex determined by its all odd or even vertUsually, we denote one orientation of s by s denoted by s = a0a1 ·are a0, a1, · · ·, an formally, and another by −s = −a0a1 · · ·an in t

Denition 3.1.6 Let K be a simplicial complex with an orientaq-dimensional simplexes in K , where q > 0, an integer. A q-dim


Denition 3.1.7 A boundary homomorphism ∂ q : C q(K ) →s = a0a1, · · ·aq is dened by

∂ s =q

( 1)ia0a1 a i a



∂ qs =i=0

(−1) a0a1 · · · a i · · ·aq,

where

a i means delete the vertex a i and extending it to ∀c∈

for c =

α q

i=1 c(s i)s i∈C q(K ),

∂ q(c) =α q

i=1

c(s i )∂ q(s i)

and ∂ q(c) = 0 if q ≤0 or q > dimK .

For example, we know that ∂ 1a0a1 = a1

−a0 and ∂ 2a0a1a

a0a1 + a1a2 + a2a0 for simplexes in Fig.3.1.4.

¹

¹

«

a0 a1

a0

a1

Fig. 3.1.4

These boundary homomorphisms ∂ q have an important next result, which brings about the conception of chain com

Theorem 3.1.14 ∂ q− 1∂ q = 0 for ∀q∈Z .

Proof We only need to prove that ∂q 1∂q = 0 for∀s∈Tq

106 Chap.3

+q

j = i+1

(

−1) j − 1a0a1

· · · a i

· · · a j

· · ·aq

=0≤ j<i ≤ q

(−1)i+ j a0a1 · · ·a j · · ·a i · · ·aq



−0≤ i<j ≤ q

(−1)i+ j a0a1 · · · a i · · · a j · · ·aq

= 0 .

This completes the proof.A chain complex (C ; ∂ ) is a sequence of Abelian groups and

0 → · · · →C q+1∂ q+1

→C q∂ q

→C q− 1 → · · · →0such that ∂ q∂ q+1 = 0 for ∀q∈Z. Whence, Im∂ q+1

⊂Ker∂ q in(C ; ∂ ).

By Theorem 3.1.14, we know that chain groups C q(K ) with∂ q on a simplicial complex K is a chain complex

0 → · · · →C q+1 (K )∂ q +1

→C q(K )∂ q

→C q− 1(K ) → · ·The simplicial homology group is dened in the next.

Denition 3.1.8 Let K be an oriented simplicial complex with a

0 → · · · →C q+1 (K )

∂ q +1

→C q(K )

∂ q

→C q− 1(K ) → · ·Then Z q(K ) = Ker ∂ q, Bq(K ) = Im ∂ q+1 and H q = Z q(K )/B q(K ) a

f i li i l l th g f i li i l b d i d


a



b c

d

Fig. 3.1.5

In this planar graph, abc, abd, acd and bcd are 2-simplexdene their orientations to be a →b →c →a, a →b →d →b →c →d →b. Then c = abc −abd + acd −bcd is a 2-cycle

∂ 2c = ∂ 2(abc) −∂ 2(abd) + ∂ 2(acd) −∂ 2(bcd)

= bc−ac + ab −bd + ad −ab + cd −ad + ac

= 0 .

Denition 3.1.9 Let K be an oriented simplicial complexwith αq q-dimensional simplexes, where q = 0 , 1, · · ·, dimKcharacteristic χ (K ) of K is dened by

χ (K ) =

dim K

q=0(−1)qαq.

For example, the Euler -Poincare characteristic of 2-com

108 Chap.3

0 → · · · →C q+1 (K )∂ q +1

→C q(K )∂ q

→C q− 1(K ) → · ·Notice that each C q(K ) is a free Abelian group of rank α

H = Z (K )/B (K ) = Ker ∂ / Im∂ Then



H q = Z q(K )/B q(K ) = Ker ∂ q/ Im∂ q+1 . Then

rank H q(K ) = rank Z q(K ) −rank Bq(K ).

In fact, each basis {B1, B2, · · ·, B rank B q (K )}of Bq(K ) can be ex

{Z 1, Z 2, · · ·, Z rank Z q (K )}by adding a basis {H 1, H 2, · · ·, H rank H q (K )

Applying Corollary 2 .2.3, we get that Bq− 1(K )∼= C q(K )/Z q

rank Bq− 1(K ) = α q

−rank Z q(K )

Notice that rank B− 1(K ) = rank Bdim K = 0 by denition, we

χ (K ) =dim K

q=0

(−1)qαq

=dim K

q=0

(−1)q(rank Z q(K ) + rank Bq− 1(K )

=dim K

q=0

(−1)q(rank Z q(K ) −rank Bq(K ))

=

dim K

q=0 (−1)q

rank H q(K ).


Classifying n-manifolds for a given integer n is an impor

object in topology. However, if n = 2, i.e., the classication for details), particularly for surfaces, i.e., 2-connected manifo

For classifying surfaces, T.Rad´o presented a combinatori



that there exists a triangulation {T i , i ≥1}on any surface S .one to dene a surface combinatorially, i.e., a surface is top

polygon with even number of edges by identifying each pairsdirection on it. If label each pair of edges by a letter e, eidentifying with a cyclic permutation such that each edge etimes in S , one is e and another is e− 1. Let a,b,c, · · ·denoA,B,C, · · ·the sections of successive letters in a linear ordstring of letters on S ). Then, a surface can be represented as

S = ( · · ·,A,a ,B,a − 1, C, · · ·),where, a ∈ E ,A,B,C denote a string of letters. Dene thremations as follows:

(O1) (A,a,a − 1, B )

⇔

(A, B );

(O2) (i) (A,a,b,B,b− 1, a− 1)⇔(A,c,B,c− 1);

(ii ) (A,a,b,B,a,b )⇔(A,c,B,c);

(O3) (i) (A,a,B,C,a − 1, D )⇔(B,a,A,D,a − 1, C )

(ii ) (A,a,B,C,a,D )⇔(B,a,A,C − 1, a ,D − 1

If a surface S can be obtained from S 0 by these elemO1-O3, we say that S is elementary equivalent with S 0, deno

110 Chap.3

a1a1a2a2 · · ·an an .

Proof By operations O1 −O3, we can prove that

AaBbCa − 1Db− 1E ∼El ADCBEaba − 1b− 1,



AcBcC ∼El AB − 1cc,

Accaba− 1b− 1

∼

El Accaabb.

Applying the inductive method on the cardinality of E , we get th

Now let S be a topological space with a collection C of openequivalence on points in S . For convenience, denote C [u] = {vS/ ∼S = {C [u]|u∈S }. There is a natural mapping p form S toby p(u) = [u], similar to these covering spaces.

We dene a set U in S/ ∼S to be open if p− 1(U ) ∈S is othese open sets in S/ ∼s , S/ ∼S become a topological space, cspace of S under ∼S .

For example, the combinatorial denition of surface is just an aquotient space, i.e., a polygon S with even number of edges und

∼S on pairs of edges along a given direction. Some well-known susphere, the torus and Klein Bottle, are shown in Fig.3 .1.6.

¹

¹

¹

¹

¹

sphere torus projective plane Kl

Sec.3.2 Euclidean Geometry

and

H q(P n ) =⎧⎪⎪⎪⎨⎪⎪⎪⎩

Z, q = 0,2n

Z⊕Z⊕ · · ·⊕Z, q =0 0 1



⎪⎨⎪⎪⎪⎩ 0, q = 0 , 1

H q(Qn ) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

Z, q n − 1

Z⊕Z⊕ · · ·⊕Z⊕Z2, q0, q =

for any integer n ≥0.

§3.2 EUCLIDEAN GEOMETRY

3.2.1 Euclidean Space. An Euclidean space on a real vecF is a mapping

·, · : E ×E →R with (e1, e2) →e1, e2 for∀such that for e, e1, e2∈E , α∈

F

(E1) e, e1 + e2 = e, e1 + e, e2 ;(E2) e,αe 1 = α e, e1 ;(E3) e

1, e

2= e

2, e

1;

(E4) e, e ≥0 and e, e = 0 if and only if e = 0.

In a Euclidean space E the number e e is called it

112 Chap.3

For ( ii ), applying (E1)-(E2), we know that

n

i=1

xie1i ,

m

j =1

yie2 j =

m

j =1

n

i=1

xi e1i , yie2

j



i 1 j 1 j 1 i 1

=m

j =1

yi

n

i=1

xi e1i , e2

j

=m

j =1

yi e2 j ,

n

i=1

xie1i

=n

i=1

m

j =1

xiyi e2 j , e1

i

=

n

i=1

m

j =1xiyi e

1i , e

2 j .

Theorem 3.2.1 Let E be a Euclidean space. Then for ∀e1, e2∈(i) | e1, e2 | ≤e1 e2 ;

(ii ) e1 + e2 ≤e1 + e2 .

Proof Notice that the inequality ( i) is hold if e1 or e2 = 0. Ax = e1 ,e2

e1 ,e1. Since

e2 −xe1, e2 −xe1 = e2, e2 −2 e1, e2 x + e1, e1 x

Replacing x by e1 ,e2e1 ,e1

in it, we nd that

e1, e1 e2, e2 −e1, e22 ≥0.

Th f h


≤ e1, e1 + 2 e1, e1 e2, e1

= ( e1 + e2 )2.

Whence,



e1 + e2 ≤e1 + e2 .

Denition 3.2.1 Let E be a Euclidean space, a, b∈E , abetween a and b are determined by

cos θ =a, b

a b

.

Notice that by Theorem 3 .2.1(i), we always have that

−1 ≤a, b

a b ≤ −1.

Whence, the angle between a and b is well-dened.Denition 3.2.2 Let E be a Euclidean space, x, y∈E .if x, y = 0 . If there is a basis e1, e2, · · ·, em of E suchorthogonal two by two, then this basis is called an orthogonaei = 1 for 1 ≤ i ≤m, an orthogonal basis e1, e2, · · ·, em is

Theorem 3.2.2 Any n-dimensional Euclidean space E has

Proof Let a1, a2, · · ·, an be a basis of E . We constru

114 Chap.3

Assume we have constructed b1, b2, · · ·, bk for an integer 1

each of which is a linear combination of a1, a2, · · ·, a i , 1 ≤ ib1, b1 , b2, b2 , · · ·, bk− 1, bk− 1 = 0. Let

ak b1 ak b2 ak bk 1



bk = ak −ak , b1

b1, b1b1 −

ak , b2

b2, b2b2 −· · ·−

ak , bk− 1

bk− 1, bk− 1

Then bk is a linear combination of a1, a2, · · ·, ak− 1 and

bk , bi = ak , bi −ak , b1

b1, b1b1, bi −· · ·−

ak , bk− 1

bk− 1, bk− 1

= ak , bi

−

ak , bi

bi , bi

bi , bi = 0

for i = 1 , 2, · · ·, k −1. Apply the induction principle, this proof is

Corollary 3.2.1 Any n-dimensional Euclidean space E has a no

Proof According to Theorem 3 .2.2, any n-dimensional Eucli

an orthogonal basis a1, a2, · · ·, am . Now let e1 =a 1a 1 , e2 =

a 2a 2

Then we nd that

ei , e j =a i , a j

a i a j= 0

and

ei =a i

a=

a i

a= 1


Theorem 3.2.3 Two nite dimensional Euclidean spaces E

and only if dimE 1 = dim E 2.Proof By Denition 3.2.3, we get dimE 1 = dim E 2 if ENow if dimE 1 = dim E 2, we prove that they are isomorp



dimE 2 = n. Applying Corollary 3 .2.1, choose normal basesb1, b2, · · ·, bn of E 2, respectively. Dene a 1 −1 mapping h :

for 1 ≤ i ≤n and extend it linearity on E 1, we know that

h(n

i=1

xia i) =n

i=1

xi h(a i).

Letn

i=1xia i and

n

i=1yia i be two elements in E 1. Then w

n

i=1

xia i ,n

i=1

yia i =n

i=1

xiyi

and

h(

n

i=1xia i), h(

n

i=1yia i) =

n

i=1xi yi


n

i=1

xia i ,n

i=1

yia i = h(n

i=1

xi a i), h(n

i=1

Notice that R n is an n-dimensional space with a normal

116 Chap.3

T (αa + b) = αT (a) + T (b)

for∀a, b∈E 1 and ∀α∈F 1.

If F 1 = F 2 = R , all such linear mappings T from E 1 to E 2 fo



over R , denoted by L(E 1, E 2). It is obvious that L(E 1, E 2)⊂E E2

Theorem 3.2.4 If dimE 1 = n, dimE 2 = m, then dim L(E 1, E 2) =Proof Let e1

1, e12, · · ·, e1

n and e21, e2

2, · · ·, e2m be basis of E 1 and

For each pair ( i, j ), 1 ≤ i ≤n, 1 ≤ j ≤m, dene an element lij

lij (e1i ) = e2

j and lij (e1k ) = 0 if k = i.

Then for x = n

i=1xie1

i ∈E 1, we have lij (x) = xi e2 j . We prove th

1 ≤ j ≤m consists of a basis of L(E 1, E 2).In fact, if there are numbers xij ∈R , 1 ≤ i ≤n, 1 ≤ j ≤m

n

i=1

m

j =1

xij lij = 0 ,

then

n

i=1

m

j =1

xij lij (e1i ) = 0( e1

i ) = 0

for e1i , 1 ≤ i ≤n. Whence, we nd that

m


f (e1k) =

m

j =1μkj e2 j =

n

i=1

m

j =1μij lij (e1k)

By the linearity of f , we get that



f =m

j =1

μkj e2 j =

n

i=1

m

j =1

μij lij ,

i.e., f is linearly spanned by lij , 1 ≤ i ≤n, 1 ≤ j ≤m.Consequently, dim L(E 1, E 2) = nm .

In L(E , E 1), if E 1 = R , the linear space L(E , R ) consif : E →R , is called the dual space of E , denoted by E∗.

3.2.4, we get the next consequence.Corollary 3.2.3 dimE∗= dim E .

Now let E 1, E 2, · · ·, E k and F be linear spaces over eF , respectively, a mapping

T : E 1 ×E 2 ×· · ·×E k →F

is called k-multilinear if T is linear in each argument separat

T (e1, · · ·, αe i + βf i , · · ·, ek) = αT (e1, · · ·, ei , · · ·, ek) + β

for α, β

∈

F i , 1

≤i

≤k. All such multilinear mappings als

denoted by L(E 1, E 2, · · ·, E k; F ). Particularly, if E i = E fordenoted by Lk(E , F ).

118 Chap.3

u1⊗ · · ·⊗u p⊗v∗1⊗ · · ·⊗v∗q(x∗1, · · ·, x∗ p, y1, · · ·, yq) = x∗1(u1) · · ·x∗ p(u

Let e1, · · ·, en be a basis of E and e∗1, · · ·, e∗n of its dual E∗

Th 3 2 4 k h T∈T p q(E ) b i l



Theorem 3 .2.4, we know that any T ∈T p,q(E ) can be uniquely w

T =i1 ,··· ,i p ,j 1 ,··· ,j q

T i1 ,··· ,i p

j1 ,··· ,j q ei1

⊗ · · ·⊗eip

⊗e∗ j1

⊗ · · ·for components T i1 ,··· ,i p

j 1 ,··· ,j q ∈R .

3.2.3 Differential Calculus on R n . Let R n , R m be Euclideopened set U ⊂R n , let f : U →R m be a mapping from U into R

f (x1, x2, · · ·, xn ) = ( f 1(x1, x2, · · ·, xn ), f 2(x1, x2, · · ·, xn ), · · ·, f m

also written it by f = ( f 1, f 2, · · ·, f m ) for abbreviation. Thendifferentiable at a point x∈U if there exists a linear mapping Athat

f (x + h) = f (x) + Ah + r (h)

with r : U →R m ,

limh→0

r (h)h

= 0

for all h∈R n with x + h∈U hold. This linear mapping A is calof f at x∈U , denoted by


d(fg )(x) = f (x)dg(x) + g(x)df (x);

d(λx ) = λdf (x),

where λ∈R .A map f : U ⊂R n →R m is said to have n partial deriv



D i f (x) = limt→0

f (x + t i) −f (x)

t=

df (x + t i)

dt |t=0

at x∈U , if all these n mappings gi(t) = f (x + t i ) are diffusually denote the D i f (x) by ∂f

∂x i(x).

Theorem 3.2.5 Let f : U ⊂R n →R m be a differentiable mof the differential df (x) with respect to the normal bases of

(A ji ) =⎛⎜⎜⎝

∂f 1

∂x 1(x) · · ·∂f 1

∂x n(x)

......

∂f m

∂x 1(x) · · ·∂f m

∂x n(x)⎞⎟⎟⎠

= (∂f j

∂x i(x)), 1 ≤ i

which is referred to as the Jacobian matrix and its determ

Jacobian of f at the point x∈U , usually denoted by

∂ (f 1, · · ·, f m )∂ (x1 , · · ·, xn )

= det(∂f j

∂x i(x)) .

Proof Let x = ( x1, · · ·, xn )∈U ⊂R n , x + h = ( x1 +

Then for such h,

n

120 Chap.3

∂f j

∂x i (x1, · · ·, xn ) = A ji

for h i →0.

Corollary 3.2.4 Let f : U ⊂R n →V ⊂R m and g : V →R



mappings. Then the composite mapping h = gf : U →R p is also its differential, the chain rule.

dg(x) = dg(f (x))df (x).

Proof Not loss of generality, let f = ( f 1, · · ·, f m ) and gdifferentiable at x∈U , y = f (x) and h = ( h1, · · ·, h p), respectivchain rule on hk = gk(f 1,

· · ·, f m ), 1

≤k

≤ p in one variable, we

∂h k

∂x i=

m

j =1

∂gk

∂y j

∂f j

∂x i.

Choose the normal bases of R n , R m and R p. Then by Theorethat

dh(x) =⎛⎜⎜⎝

∂h 1∂x 1

(x) · · ·∂h 1∂x n

(x)...

...∂f p

∂x 1(x) · · ·∂f p

∂x n(x)⎞⎟⎟⎠

=

⎛⎜⎜⎝

∂g 1

∂y 1(y) · · ·∂g 1

∂y m(y)

..

....∂g p

∂y 1(y) · · ·∂g p

∂y m(y)⎞⎟⎟⎠

×⎛⎜⎜⎝

∂f 1

∂x 1(x) · · ·∂f 1

∂x n(x)

..

.∂f m

∂x 1(x) · · ·∂f m

∂x n(x)

dg(f ( ))df ( )


A bijective mapping f : U →V , where U, V ⊂R n , is

f ∈C k(U,R

n) and f

− 1

∈C k(V, R

n). Certainly, a C

k-diff

also a homeomorphism.For determining a C k-diffeomorphism mapping, the foll

theorem is usually applicable. Its proof can be found in, for



Theorem 3.2.6 Let U be an open subset of R n ×R m and

of class C k , 1 ≤ k ≤ ∞. If f (x0, y0) = 0 at the point (x0, ymatrix ∂f j /∂y i(x0, y0) is non-singular, i.e.,

det(∂f j

∂y i (x0, y0)) = 0 , where 1 ≤ i, j ≤Then there exist opened neighborhoods V of x0 in R n and W

mapping g : V →W such that V ×W ⊂U and for each (x,

f (x, y) = 0⇒y = g(x).

3.2.4 Differential Form. Let R n be an Euclidean spa

1,

2,

· · ·,

n. Then

∀

x

∈

R n , there is a unique n-tuple ( x1, x

that

x = x1 1 + x2 2 + · · ·+ xn n .

For needing in research tangent spaces of differential ma

chapters, we consider a vector space

G(Λ) = Λ0⊕Λ1⊕Λ2⊕ · · ·⊕Λn

122 Chap.3

Notice that dx i1∧dx i2 = −dxi2∧dxi1 by the denition of ∧. G

in Λk

, 1 ≤k ≤n, have a form

i1 <i 2 < ···<i k

a i1 i2 ··· ik (x1, x2, · · ·, xn )dxi1∧dx i2∧ · · ·dff

k



A differential k-form is an element in Λk for 1 ≤ k ≤ n. ItC ∞ if each function a i1 i2 ··· ik (x1, x2,

· · ·, xn ) is of class C ∞ . By de

in G(Λ) can be represented as

a(x1, x2, · · ·, xn ) +n

i=1

a i(x1, x2, · · ·, xn )dxi

+n

i1 <i 2

a i1 i2 (x1, x2,

· · ·, xn )dxi1

∧

dx i2 +

· · ·+

n

i1 <i 2 < ···<i k

a i1 i2 ··· ik (x1, x2, · · ·, xn )dxi1∧dxi2∧ · · ·+ a1,2,··· ,n (x1, x2, · · ·, xn )dx1∧dx2∧ · · ·∧dxn .

An exterior differential operator d : Λk

→Λk+1 is dened by

dω =i1 <i 2 < ···<i k i=1

(∂a i1 i2 ··· ik

∂x idxi )∧dx i1∧ · · ·∧dx

for a differential k-form

ω =i1 <i 2 < ···<i k

a i1 i2 ··· ik (x1, x2, · · ·, xn )dx i1∧dx i2∧ · · ·d

A diff ti l f i ll d t b l d if d 0 d



ddω =n

i=1

d(∂a∂x i

)dxi∧dxi1∧ · · ·∧dxik

n ∂2a



=i,j =1

∂ a∂x i ∂x j

dx j∧dx i∧dxi1∧ · · ·∧dxik

=n

i<j

∂ 2a∂x i ∂x j

(dxi∧dx j + dx j∧dxi)∧dx i

= 0

3.2.5 Stokes’ Theorem on Simplicial Complex. A sR p is dened by

s p = {(x1, · · ·, x p)∈R p| p

i=1

xi ≤1, 0 ≤xi ≤1 fo

Now let ω∈Λ p

be a differential p-form with

ω =i1 <i 2 < ···<i p

a i1 i2 ··· ip (x1, x2, · · ·, xn )dxi1∧dxi2

Its integral on sn is dened by

s

ω =i1 <i 2 < ···<i p

· · · a i1 i2 ··· ip (x1, x2, · · ·, xn )dx

124 Chap.3

∂c p

ω = cpdω.

Proof By denition, it is suffices to check that



∂s p

ω = sp

dω

in the case of ω being a monomial, i.e.,

ω = a(x)dx1∧ · · ·∧d x j∧ · · ·∧dx p

with a xed j, 1 ≤ j ≤ p on a p-simplex s p = a0a1 · · ·a p. Then w

sp

dω = sp

( p

i=1

∂a∂x i

dxi)∧dx1∧ · · ·∧d x j∧ · ·

= ( −1) j − 1 sp

∂a∂x i

dx1∧ · · ·∧dx p

= ( −1) j − 1

a ( j )p − 1

[a(B) −a(A)]dx1 · · ·d x j · · ·d

where a( j ) p− 1 is a ( p −1)-simplex determined by a( j )

p− 1(x1, · · ·, x j

a(x1, · · ·, x j − 1, 0, · · ·, x p) and a(B) = a(x1, · · ·, x j − 1, 1 − (x1 +x p),

· · ·, x p), see Fig.3.2.1 for details.

x j

a


s p

dω = ( −1) j a ( j )p − 1

a(A)dx1 · · ·d x j · · ·dx p + ( −1) j − 1

a ( j )p − 1

= ( −1) j ( j )

ω + ( −1) j − 1 ( j )

a(B)dx1 · · ·dx j · · ·dx



a ( j )p − 1 a ( j )

p − 1 Let τ be a mapping τ : a0

→a j and a i

→a i if i = j , w

on coordinates ( x1, x2, · · ·, x p) →(x j , x1, · · ·, x j , · · ·, x p). Wh

a (0)

p − 1

ω = a ( j )

p − 1

a(B)∂ (x1, x2, · · ·, x p)

∂ (x j , x1, · · ·, x j , · · ·, x p)dx1 ·

= ( −1) j − 1

a ( j )p − 1

a(B)dx1 · · ·d x j · · ·dx p.

Notice that if i = 0 or j , then

a

( i )p − 1

ω = 0

Whence, we nd that

(−1) j a ( j )

p − 1

ω + ( −1) j − 1(−1) j − 1 a (0)

p − 1

ω = p

i=0

(−

and

ω ω p

( 1)i ω



Sec.3.3 Smarandache manifolds

denying some axioms in Euclidean geometry done as in Loba

geometry and Riemannian geometry.For example, let R 2 be a Euclidean plane, points A, B

line, where each straight line passes through A will turn 3and passes through B will turn 30o degree to the down such3 3 1 Th h li i h h A i F ill i



3.3.1. Then each line passing through A in F 1 will intersethrough B in F 2 will not intersect with l and there is only oother points does not intersect with l.

¹

¹

A

B

l30o

30o

30o

30o

.........................

...... ..........................

...................

.......................

F 1

F 2

Fig. 3.3.1

A nice model on Smarandache geometries, namely s-mafound by Iseri in [Ise1], which is dened as follows:

An s-manifold is any collection C (T, n) of these equT i , 1 ≤ i ≤n satisfying the following conditions:

(i) each edge e is the identication of at most two edgtriangular disks T i , T j , 1 ≤ i, j ≤n and i = j ;

(ii ) each vertex v is the identication of one vertex in e

distinct triangular disks.

The vertices are classied by the number of the disks a

128 Chap.3

In a plane, an elliptic vertex O, a Euclidean vertex P and

tex Q and an s-line L1, L2 or L3 passes through points O, P oFig.3.3.2(a), (b), (c), respectively.

As shown in [Ise1] and [Mao3], there are many ways for constdache geometry, such as those of denial one or more axioms of a Eb



by new axiom or its anti-axiom,..., etc.

3.3.2 Map Geometry. A map geometry is gotten by endfunction μ : V (M ) →[0, 4π) on a map M , which was rst introda generalization of Iseri’s model on surfaces. In fact, the essencis not these numbers 5 , 6 or 7, but in these angles 300o, 360o andwhich determines a vertex is elliptic, Euclidean or hyperbolic on t

Denition 3.3.1 Let M be a combinatorial map on a surface Svalency ≥ 3 and μ : V (M ) →[0, 4π), i.e., endow each vertex ua real number μ(u), 0 < μ (u) < 4π

ρM (u) . The pair (M, μ ) is callewithout boundary, μ(u) an angle factor on u and orientable or nois orientable or not.

Certainly, a vertex u ∈V (M ) with ρM (u)μ(u) < 2π, = 2realized in a Euclidean space R 3, such as those shown in Fig.3.3.

u

u

u


to 180o only when u is Euclidean. For convenience, we alw

passing through an elliptic point turn to the left and a hyperbon the plane.

Theorem 3.3.1 Let M be a map on a locally orientable suρM (u) ≥ 3 for ∀u∈V (M ). Then there exists an angle fact



≥ ∀∈such that (M, μ ) is a Smarandache geometry by denial the a

(E5),(L5) and (R5).Proof By the assumption ρM (u) ≥3, we can always c

such that μ(u)ρM (u) < 2π, μ(v)ρM (u) = 2π or μ(w)ρM (u)u,v,w∈V (M ), i.e., there elliptic, or Euclidean, or hyperbolisimultaneously. The proof is divided into three cases.

Case 1. M is a planar map

Choose L being a line under the map M , not intersectiThen if u is Euclidean, there is one and only one line passinsecting with L, and if u is elliptic, there are innite many lnot intersecting with L, but if u is hyperbolic, then each line

intersect with L. See for example, Fig.3.3.4 in where the plagraph K 4 on a sphere and points 1 , 2 are elliptic, 3 is Eucis hyperbolic. Then all lines in the eld A do not intersectpassing through the point 4 will intersect with the line L. Smarandache geometry by denial the axiom (E5) with these a

(R5).¿

130 Chap.3

According to Theorem 3 .1.15 of classifying surfaces, We only

assertion on a torus. In this case, lines on a torus has the follow[NiS1] for details):

if the slope ς of a line L is a rational number, then L is a torus. Otherwise, L is innite, and moreover L passes arbitrar



point on the torus.

Whence, if L1 is a line on a torus with an irrational slope not paelliptic or a hyperbolic point, then for any point u exterior to L1, point, then there is only one line passing through u not intersectinu is elliptic or hyperbolic, any m-line passing through u will inte

Now let L2 be a line on the torus with a rational slope not pa

elliptic or a hyperbolic point, such as the the line L2 shown in Fv is a Euclidean point. If u is a Euclidean point, then each lineu with rational slope in the area A will not intersect with L2, butthrough u with irrational slope in the area A will intersect with L

¹

¹

¿

¿

½

½

1

1

2 2v

uL2

L

A

Fig 3 3 5


u

1

1

2

2

22’

2’

Ã

2

2’(a) (b) (c

2



Fig. 3.3.6Let L be a line passing through the center of the circle. T

point, there is only one line passing through u such as the cais an elliptic point then there is an m-line passing through iL such as the case (b) in Fig.3.3.7. We assume the point

there exists a line passing through 1 and 0, then any line in thpassing through v will intersect with L.

¹ ¹

¹

L0 0

1

1 2

2u

¹

.......

........

..

.............

..........

...

...........

.. ............

.

........

0 01 L

vL1

22

0

L2

(a) (b)

Fig. 3.3.7

If w is a Euclidean point and there is a line passing throwith L such as the case (c) in Fig.3.3.7, then any line in thpassing through w will not intersect with L. Since the poa map M on a projective plane can be choose as our wish

132 Chap.3

(S (M ) \{f 1, f 2, · · ·, f l}, μ) is called a map geometry with boundary

orientable or not if (M, μ ) is orientable or not, where S (M ) denosurface of M .

Similarly, map geometries with boundary can also provide ometries, which is convinced in the following for l = 1.



Theorem 3.3.2 Let M be a map on a locally orientable surface witvalency ≥3 and a face f ∈F (M ). Then there is an angle factor μsuch that (M, μ )− 1 is a Smarandache geometry by denial the axioaxioms (E5),(L5) and (R5).

Proof Divide the discussion into planar map, orientable manon-orientable map on a projective plane dependent on M , resto the proof of Theorem 3.3.1, We can prove (M, μ )− 1 is a Smarby denial the axiom (E5) with these axioms (E5),(L5) and (R5) fact, the proof applies here, only need to note that a line in a maboundary is terminated at its boundary.

A Poincare’s model for hyperbolic geometry is an upper half-plare upper half-circles with center on the x-axis or upper straight lito the x-axis such as those shown in Fig.3 .3.8.

L1 L2

L4 L5

L6


L1

L2

L3



Fig. 3.3.9

Whence, a Klein’s model is nothing but a map geometface determined by Theorem 3 .3.2. This fact convinces us thaboundary are a generalization of hyperbolic geometry.

3.3.3 Pseudo-Euclidean Space. Let R n be an n-dimen

with a normal basis 1 = (1 , 0, · · ·, 0), 2 = (0 , 1, · · ·, 0), · ·orientation −→X is a vector −−→OX with −−→OX = 1 in R n , wUsually, an orientation −→X is denoted by its projections of i ≤n, i.e.,

−→X = (cos( −−→OX, 1), cos(−−→OX, 2),

· · ·, cos(−−OX

where (−−→OX, i) denotes the angle between vectors −−→OX and i

orientations −→X in R n consist of a set O .

A pseudo-Euclidean space is a pair (R n , ω|−→O ), where

continuous function, i.e., a straight line with an orientation −→O

−→O + ω

|−→O(u) after it passing through a point u

∈

E . It is obvinamely the Euclidean space itself if and only if ω|−→O (u) = 0

We have known that a straight line L passing through




on points in R n with −→V eu ∩−→V el = ∅, −→V eu ∩−→V hy = ∅and −→V

−→V el ∩−→V hy are called non-Euclidean points .Now we introduce a linear order ≺on O by the diction

following.

For (x1, x2, · · ·, xn ) and (x1, x2, · · ·, xn ) ∈O , if x1 = x

and x < x for any integer l 0 l n 1 then d



and xl+1 < x l+1 for any integer l, 0 ≤ l ≤ n −1, then d

(x1, x2, · · ·, xn ).By this denition, we know that

ω|−→O (u)≺ω|−→O (v)≺ω|−→O (w)

for∀u∈−→V el , v∈−→V eu , w∈−→V hy and a given orientation −→O .

nd an interesting result following.Theorem 3.3.3 For any orientation −→O ∈O in a pseudo-Euc

if −→V el = ∅and −→V hy = ∅, then −→V eu = ∅.

Proof By assumption, −→V el = ∅and −→V hy = ∅, we can chw

∈

−→V hy . Notice that ω

|−→O: R n

→O is a continuous and

set. Applying the generalized intermediate value theorem oin topology, i.e.,

Let f : X →Y be a continuous mapping with X a colinear ordered set in the order topology. If a, b∈X and y∈f (b), then there exists x

∈

X such that f (x) = y.

we know that there is a point v∈R n such that

136 Chap.3

if −→V eu , −→V el = ∅, or −→V eu , −→V hy = ∅, or −→V el , −→V hy = ∅for an

(Rn

, ω|−→O ).Proof Notice that ω|−→O (u) = 0 is an axiom in R n , but a Sm

axiom if −→V eu , −→V el = ∅, or −→V eu , −→V hy = ∅, or −→V el , −→V hy = ∅for in (R n , ω|−→O ) for ω|−→O (u) = 0 or = 0 in the former two cases an

0 b th h ld i th l t Wh k th t ( R n | )



0 both hold in the last one. Whence, we know that ( R n , ω|−→O )

geometry by denition.Notice that there innite points on a segment of a straight line

a necessary for the existence of a straight line is there exist innitein (R n , ω|−→O ). We nd a necessary and sufficient result for the exC in (R n , ω|−→O ) following.

Theorem 3.3.5 A curve C = ( f 1(t), f 2(t), · · ·, f n (t)) exists in a space (R n , ω|−→O ) for an orientation −→O if and only if

df 1(t)dt |u = (

1ω1(u) )

2

−1,

df 2(t)dt |u = (

1ω2(u)

)2 −1,


.............................

..........................

X

3

ω2

ω3θ3

θ1 θ2



¹

·

........

...................................................... ...............

......O

A

1

ω1

ω22

Fig. 3.3.10

Then we know that

cos θi = ωi , 1 ≤ i ≤n.

According to the geometrical implication of differentiseeing also Fig.3.3.10, we know that

df i(t)dt |u = tgθi = (

1ωi(u)

)2 −1

for 1 ≤ i ≤n. Therefore, if a curve C = ( f 1(t), f 2(t), · · ·, f nEuclidean space ( R n , ω|−→O ) for an orientation −→O , then

df i(t)dt |u = (

1ω2(u)

)2 −1, 1 ≤ i ≤n

138 Chap.3

3.3.4 Smarandache manifold. For an integer n, n ≥ 2, a Sifold is a n-manifold that supports a Smarandache geometry. Cermany ways for construction of Smarandache manifolds. For exampEuclidean spaces ( R n , ω|−→O ) for different homomorphisms ω−→O anWe consider a general family of Smarandache manifolds, i.e., (M n , Aω) in this section, which is a generalization of n-manifolds



An n-dimensional pseudo-manifold (M n ,

Aω) is a Hausdorff

each points p has an open neighborhood U p homomorphic to a space (R n , ω|−→O ), where A= {(U p,ϕω

p )| p∈M n}is its atlas with ϕ

ω p : U p →(R n , ω|−→O ) and a chart ( U p,ϕω

p ).

Theorem 3.3.6 For a point p∈(M n , Aω) with a local chart (Uand only if ω

|−→O( p) = 0 .

Proof For ∀ p ∈(M n , Aω), if ϕω p ( p) = ϕ p( p), then ω(ϕ p(

the denition of pseudo-Euclidean space ( R n , ω|−→O ), this can onω( p) = 0.

A point p∈(M n , Aω) is elliptic, Euclidean or hyperbolic if ω(ϕ

is elliptic, Euclidean or hyperbolic, respectively. These elliptic and also called non-Euclidean points . We get a consequence by Theor

Corollary 3.3.3 Let (M n , Aω) be a pseudo-manifold. Then ϕω p =

every point in M n is Euclidean.

Theorem 3.3.7 Let (M n ,

Aω) be an n-dimensional pseudo-ma

If there are Euclidean and non-Euclidean points simultaneously ohyperbolic points on an orientation −→O in (Up ϕp) then (M n Aω)



140 Chap.3

Particularly, we have consequences following by Theorem Euclidean spaces ( R n , ω

|−→O).

Corollary 3.3.4 For any integer n ≥2, if there are Euclidean anpoints simultaneously or two elliptic or hyperbolic points in an o(R n , ω|−→O ), then (R n , ω|−→O ) is an n-dimensional Smarandache geo

Corollary 3 3 4 partially answers an open problem in [Mao3



Corollary 3.3.4 partially answers an open problem in [Mao3

Smarandache geometries in R 3.

Corollary 3.3.5 If there are points p, q∈(R 3, ω|−→O ) such that ω|−→Oω|−→O (q) = (0 , 0, 0) or p, q are simultaneously elliptic or hyperbolic

−→O in (R 3, ω|−→O ), then (R 3, ω|−→O ) is a Smarandache geometry.

Notice that if there only nite non-Euclidean points in ( Mbased at a point p∈M n is still a loop of (M n , Aω) based at a poand vice versa. Whence, we get the fundamental groups of pseudnite non-Euclidean points.

Theorem 3.3.8 Let (M n , Aω) be a pseudo-manifold with nite non

Then π1(M n , p) = π1((M n , Aω), p)

for ∀ p∈(M n , Aω).

§3.4 DIFFERENTIAL SMARANDACHE MANIFOLDS

3 4 1 Diff i l M if ld A diff i l if ld (M n

Sec.3.4 Differential Smarandache manifolds

are C r ;

(3) Ais maximal, i.e., if (U,ϕ) is an atlas of M n

equiA, then (U,ϕ)∈A.

An n-manifold is smooth if it is endowed with a C ∞ dhas been known that the base of a tangent space T pM n of(M n , A) consisting of ∂

∂x i , 1 ≤ i ≤n for∀ p∈(M n , A). Mor



manifolds can be found in [AbM1], [MAR1], [Pet1], [Wes1]

3.4.2 Differential Smarandache manifold. For an integtial Smarandache manifold ( M n , Aω) is a Smarandache maniwith a C r differentiable structure Aand ω|−→O for an orientatdache n-manifold (M n , Aω) is also said to be a smooth Sma

pseudo-manifolds, we know their differentiable conditions folTheorem 3.4.1 A pseudo-Manifold (M n , Aω) is a C r dimanifold with an orientation −→O for an integer r ≥1 if cond

(1) There is a C r differential structure A= {(U α ,ϕα )|(2) ω

|−→Ois C r ;

(3) There are Euclidean and non-Euclidean points simultaor hyperbolic points on the orientation −→O in (U p,ϕ p) for a p

Proof The condition (1) implies that ( M n , A) is a C r

and conditions (2), (3) ensure ( M n , Aω) is a differential Sm

denitions and Theorem 3 .3.7.

3 4 3 Tangent Space on Smarandache manifold F

142 Chap.3

Denote all tangent vectors at a point p∈(M n , Aω) still by Tbiguous and dene addition ¡ + ¢ and scalar multiplication ¡

·¢ for

R and f ∈ p by

(u + v)(f ) = u(f ) + v(f ), (λu )(f ) = λ ·u(f ).

Then it can be shown immediately that T pM n is a vector spacoperations + and



operations ¡ + ¢ and ¡ ·¢ .

Let p∈(M n , Aω) and γ : (−ε, ε) →R n be a smooth curve inIn (M n , Aω), there are four possible cases for tangent vectors onsuch as those shown in Fig.3 .4.1, in where these L-L represent tan

¶

¹

p

(a)

©

p·

»

µ

(b)

×

¹

p

(c)

1

1

2

2

L

L

1

1

2

2

L

L

1

1

22LL

L

p

Fig. 3.4.1

By these positions of tangent lines at a point p on γ , we cois one tangent line at a point p on a smooth curve if and only if(M n , Aω). This result enables us to get the dimensional number ofspace T pM n at a point p∈(M n , Aω).

Theorem 3.4.2 For a point p∈(M n

, Aω

) with a local chart (Uexactly s Euclidean directions along i1 , i2 , · · ·, is for p, then the di

Sec.3.4 Differential Smarandache manifolds

{ ∂ ∂x i j | p | 1 ≤ j ≤s} {∂

−

∂x l , ∂ +

∂x l | p | 1 ≤ l ≤n and l = i j

is a basis of T pM n . For ∀f ∈ p, since f is smooth, we know

f ( ) f ( )n

( 0)∂ i f

( )



f (x) = f ( p) +i=1

(xi

−x0

i )∂x i

( p)

+n

i,j =1

(xi −x0i )(x j −x0

j )∂ i f ∂x i

∂ j f ∂x j

+

for∀x = ( x1, x2, · · ·, xn )∈ϕ p(U p) by the Taylor formula inR i,j, ··· ,k contains ( xi

−x0

i )(x j

−x0

j )

· · ·(xk

−x0

k), l

∈ {+ ,

−}for 1 ≤ j ≤s and l should be deleted for l = i j , 1 ≤ j ≤s.Now let v∈T pM n . By the condition (1) of denition

point p∈(M n , Aω), we get that

v(f (x)) = v(f ( p)) + v(

n

i=1(xi −x

0i )

∂ i f ∂x i ( p))

+ v(n

i,j =1

(xi −x0i )(x j −x0

j )∂ i f ∂x i

∂ j f ∂x j

) +

Similarly, application of the condition (2) in denition

point p∈(M n

, Aω

) shows thatn ∂ i f

144 Chap.3

v(f (x)) =

n

i=1 v(xi )∂ i f ∂x i ( p) =

n

i=1 v(xi)∂ i

∂x i | p(f ). (3

The formula (3 .4.2) shows that any tangent vector v in T pMby elements in the set (3 .4.1).

All elements in the set (3 .4.1) are linearly independent. Othernumbers a1, a2, , a s , a+

1 , a−1 , a+

2 , a−2 , , a+

n s , a−n s such that



numbers a , a ,

· · ·, a , a1 , a1 , a2 , a2 ,

· · ·, an − s , an − s such that

s

j =1

a i j

∂ ∂x i j

+i= i1 ,i 2 ,··· ,i s ,1≤ i≤ n

a ii

∂ i

∂x i | p = 0 ,

where i∈ {+ , −}, then we get that

a i j = (s

j =1a i j ∂ ∂x i j

+i= i1 ,i 2 ,··· ,i s ,1≤ i≤ n

a ii ∂ i

∂x i)(xi j ) =

for 1 ≤ j ≤s and

a ii = (

s

j =1

a i j

∂ ∂x i j

+i= i1 ,i 2 ,··· ,i s ,1≤ i≤ n

a ii

∂ i

∂x i)(xi) =

for i = i1, i2, · · ·, is , 1 ≤ i ≤n. Therefore, vectors in the set (3 .4.tangent vector space T pM n at the point p∈(M n , Aω).

Notice that dim T pM n = n in Theorem 3.4.2 if and only if aare Euclidean along 1, 2, · · ·, n . We get a consequence by Theor

Corollary 3.4.1 Let (M n, A) be a smooth manifold and p∈M

n

dimTpM n = n

Sec.3.5 Pseudo-Manifold Geometry

Then, we can immediately get the result on its basis of cat a point p

∈

(M n ,

Aω) similar to Theorem 3 .4.2.

Theorem 3.4.3 For any point p∈(M n , Aω) with a localare exactly s Euclidean directions along i1 , i2 , · · ·, is for pT ∗ p M n is

dimT ∗p M n = 2n s



p

−with a basis

{dx i j | p | 1 ≤ j ≤s} {d− xl| p, d+ xl| p | 1 ≤ l ≤n and l

where

dxi

| p( ∂

∂x j | p) = δi

j and d i xi

| p( ∂ i

∂x j | p) =

for i∈ {+ , −}, 1 ≤ i ≤n.

§3.5 PSEUDO-MANIFOLD GEOMETRY

Similar to the approach in Finsler geometry, we introducethese pseudo-manifolds ( M n , Aω) following.

Denition 3.5.1 A Minkowskian norm on a vector space V isuch that

(1) F is smooth on V

\{0

}and F (v)

≥0 for

∀

v

∈

V ;

(2) F is 1-homogenous, i.e., F (λv) = λF (v) for ∀λ >

(3) f ll 0 h i bili f




(2) The map π : (P, Aω1 ) →(M, A

π (ω)0 ) is onto with π

πω1 = ω0π, and regular on spatial directions of p, i.e., if theare (ω1, ω2, · · ·, ωn ), then ωi and π(ωi) are both elliptic, or Eand |π− 1(π(ωi))| is a constant number independent of p for a

(3) For ∀x∈(M, Aπ (ω)0 ) there is an open set U with x

phism T π (ω)u : (π)− 1(U π (ω) ) →U π (ω) ×G of the form T u ( p) =

su : π− 1(Uπ (ω) ) G has the property su (pωg) = su (pω)g for



su : π (U )

→G has the property su ( p g) su ( p )g for

We know the following result for principal ber bundles

Theorem 3.5.2 Let (P,M,ω π , G) be a PFB. Then

(P,M,ω π , G) = ( P,M,π,G )

if and only if all points in pseudo-manifolds (P, Aω1 ) are Euc

Proof For ∀ p∈(P, Aω1 ), let (U p,ϕ p) be a chart at p. No

only if ϕω p = ϕ p for∀ p∈(P, Aω

1 ). According to Theorem 3 .3that all points in ( P, Aω

1 ) are Euclidean.

Denition 3.5.4 Let (P,M,ωπ, G) be a PFB with dimG

ily H = {H p| p∈(P, Aω1 ), dimH p = dim T π ( p)M }of T P is

conditions ( 1) and ( 2) following hold.

(1) For ∀ p∈(P, Aω1 ), there is a decomposition

T pP = H p V p

and the restriction π∗|H p : H p →T π ( p)M is a linear isomorph

148 Chap.3

dimV p =(dimP −dimM )(2dim P −λP ( p))

dimP .

Proof Assume these Euclidean directions of the point p beinBy denition π is regular, we know that π( 1), π ( 2), · · ·, π ( λ P ( p)) in (M, A

π (ω)1 ). Now since

π− 1(π( 1)) = π− 1(π( 2)) = = π− 1(π( λ ( ))) = μ = c



π (π( 1)) π (π( 2))

· · ·π (π( λ P ( p))) μ c

we get that λP ( p) = μλ M , where λM denotes the correspondent Euin (M, A

π (ω)1 ). Similarly, consider all directions of the point p,

dimP = μdimM . Thereafter

λM =dimM dimP λP ( p). (3.5

Now by Denition 3.5.4, T pP = H p V p, i.e.,

dimT pP = dim H p + dim V p. (3

Since π∗

|H p : H p

→T π ( p)M is a linear isomorphism, we kno

dimT π ( p)M . According to Theorem 3 .4.2, we get formulae

dimT pP = 2dim P −λP ( p)

and

dimT π ( p)M = 2dim M −λM = 2dim M −dimM dimP λP

N l i h f l i (3 5 2) h


Corollary 3.5.1 Let (P,M,ω π , G) be a P F B with a co

∀

p

∈

(P,

Aω1 ),

dimV p = dim P −dimM

if and only if the point p is Euclidean.

Now we consider conclusions included in Smarandache gin pseudo-manifold geometries.



Theorem 3.5.4 A pseudo-manifold geometry (M n ,ϕω) witon T M n is a Finsler geometry if and only if all points of (M

Proof According to Theorem 3 .3.6, ϕω p = ϕ p for∀ p∈

p is Euclidean. Whence, by denition ( M n ,ϕω) is a Finsler

all points of (M n

,ϕω

) are Euclidean.Corollary 3.5.2 There are inclusions among Smarandache gometry, Riemann geometry and Weyl geometry:

{Smarandache geometries }⊃ {pseudo-manifold

⊃ {Finsler geometry

}⊃ {Riemann geometry

}⊃Proof The rst and second inclusions are implied in ThOther inclusions are known in a textbook, such as [ChC1] an

Now let us to consider complex manifolds. Let zi = xi

complex manifold M nc is equal to a smooth real manifold M

{∂

∂x i ,∂

∂y i }for T pM nc at each point p∈M

nc . Dene a Herm

a manifold M nc endowed with a Hermite inner product h( p

150 Chap.3

κ(X, Y ) = g(X,JY ), ∀X, Y ∈T pM nc ,∀ p∈M nc

Similar to Theorem 3 .5.3 for real manifolds, we know the ne

Theorem 3.5.5 A pseudo-manifold geometry (M nc ,ϕω) with a Mon T M n is a K¨ ahler geometry if and only if F is a Hermite innewith all points of (M n ,ϕω) being Euclidean.

Proof Notice that a complex manifold M nc is equal to a re



Similar to the proof of Theorem 3 .5.3, we get the claim.

As a immediately consequence, we get the following inclusiongeometries.

Corollary 3.5.3 There are inclusions among Smarandache geometri

geometry and K¨ ahler geometry:

{Smarandache geometries }⊃ {pseudo-manifold g

⊃ {K ahler geometry

§3.6 REMARKS

3.6.1 These Smarandache geometries were proposed by Smarancontradicts axioms ( E 1) −(E 5) in a Euclid geometry, such as thgeometry, non-geometry, counter-projective geometry and anti-geoper [Sma2] for details. For example, he asked whether there existsaxioms (E 1) −(E 4) and one of the axioms following:



152 Chap.3

Corollaries 3.5.2 and 3.5.3 are interesting results established iconvince us that Smarandache geometries are indeed a generalizatialready existence. [SCF1] and other papers also mentioned thesreviewing Mao’s work.

Now we consider some well-known results in Riemannian gean orientable compact surface. Then



S Kdσ = 2πχ (S ),

where K and χ (S ) are the Gauss curvature and Euler characterformula is the well-known Gauss-Bonnet formula in differential geomThen what is its counterpart in pseudo-manifold geometries? Thiproblems following.

(1) Find a suitable denition for curvatures in pseudo-manifo(2) Find generalizations of the Gauss-Bonnet formula for pse

ometries, particularly, for pseudo-surfaces.

For an oriently compact Riemannian manifold ( M 2 p, g), let

Ω = (−1) p22 pπ p p!

i1 ,i 2 ,··· ,i 2p

δi1 ,··· ,i 2p1,··· ,2 p Ωi1 i2∧ · · ·∧Ωi2p − 1 i2p

where Ωij is the curvature form under the natural chart {ei}of M

δi1 ,··· ,i 2p

1,··· ,2 p=⎧

⎪⎪⎨⎪⎪⎩

1, if permutation i1 · · ·i2 p is even

−1, if permutation i1

· · ·i2 p is odd

0, otherwise.

Sec.3.6 Remarks

gauge elds with gravitation. In section 3 .5, we have introdmanifolds. For applying pseudo-manifolds to physics, similainduces a new gauge theory, which needs us to solving proble

to establish a gauge theory on those of pseudo-manifoldadditional conditions.

In fact, this object requires us to solve problems following:



(1) nd these conditions such that we can establish a gamanifolds;

(2) nd the Yang-Mills equation in a gauge theory on p(3) unify these gauge elds and gravitation.

CHAPTER 4.



Combinatorial Manifolds

A combinatorial manifold is a topological space consisting of mderlying a combinatorial structure, i.e., a combinatorial system of Certainly, it is a Smarandache system and a geometrical multi-spacour WORLD. For introducing this kind of geometrical spaces, wetopological behavior in this chapter, and then its differential behafollowing chapters. As a concrete introduction, Section 4 .1 presention on the dimension of combinatorial Euclidean spaces and the tion of a Euclidean space with dimension

≥4 to combinatorial Euc

with lower dimensions. This model can be also used to describe dimension≥4 in physics. The combinatorial manifold is introduce4.2. In this section, these topological properties of combinatoriasuch as those of combinatorial submanifold, vertex-edge labeled gbinatorial equivalence, homotopy class and Euler-Poincar´ e chara

etc. are discussed. Fundamental groups and singular homologycombinatorial manifolds are discussed in Sections 4 .3 and 4.4, in

Sec.4.1 Combinatorial Spaces

§4.1 COMBINATORIAL SPACES

A combinatorial space S G is a combinatorial system C G

(Σ 1;R1), (Σ 2;R2), · · ·, (Σ m ;Rm ) for an integer m with an Denition 2.1.3. We concentrated our attention on each (Σ i

space for integers i, 1 ≤ i ≤m in this section.

4.1.1 Combinatorial Euclidean Space. A combinator



combinatorial system C G of Euclidean spaces R n 1 , R n 2 , · · ·, Rstructure G, denoted by E G (n1, · · ·, nm ) and abbreviated to E

r . It is itself a Euclidean space R n c . Whence, it is natural toproblem on Euclidean spaces following.

Parking Problem Let Rn 1

, Rn 2

, · · ·, Rn m

be Euclidean spado they consist of a combinatorial Euclidean space E G (n1, ·

By our intuition, this parking problem is related with thR n 2 , · · ·, R n m , also with their combinatorial structure G. Nspace R n is an n-dimensional vector space with a normal b

2 = (0 , 1, 0 · · ·, 0), · · ·, n = (0 , · · ·, 0, 1), namely, it has n oSo if we think any Euclidean space R n is a subspace of a Eua nite but sufficiently large dimension n∞ , then two EuclR n v have a non-empty intersection if and only if they have Whence, we only need to determine the number of different o

inE

G (n1, · · ·, nm ).Denoted by X v1 , X v2 , · · ·, X vm consist of these orthogon

156 Chap.4 C

E G (n1, · · ·, n m ) of R n 1 , R n 2 , · · ·, R n m , which transfers the parkinclidean spaces to a combinatorial problem following.

Intersection Problem For given integers κ, m ≥2 and n1, n2,sets Y 1, Y 2, · · ·, Y m with their intersection graph being G such thatm, and |Y 1∪Y 2∪ · · ·∪Y m | = κ.

This enables us to nd solutions of the parking problem some

Th 4 1 1 L E ( ) b bi i l E lid



Theorem 4.1.1 Let E G (n1, · · ·, nm ) be a combinatorial Euclidean s

· · ·, R n m with an underlying structure G. Then

dimE G (n1, · · ·, nm ) =vi∈V (G) |1≤ i≤ s ∈CL s (G)

(−1)s+1 dim(R n v 1 R

where nvi denotes the dimensional number of the Euclidean space

CLs (G) consists of all complete graphs of order s in G.

Proof By denition, R n u ∩R n v = ∅only if there is an edgThis condition can be generalized to a more general situation, i.e

· · ·∩R n v l = ∅only if v1, v2, · · ·, vl G∼= K l .In fact, if R n v 1 ∩R n v 2 ∩···∩R n v l = ∅, then R n v i ∩R n v j = ∅,

(R n v i , R n v j )∈E (G) for any integers i,j, 1 ≤ i, j ≤ l. Therefore,a complete graph of order l in the intersection graph G.

Now we are needed to count these orthogonal orientations inIn fact, the number of different orthogonal orientations is

dimE G (n1,

· · ·, nm ) = dim(

v∈V (G)

R n v )

by previous discussion. Applying Theorem 1 .5.1 the inclusion-ex


Notice that dim( R n v 1 ∩R n v 2 ∩···∩R n v s ) = nv1 if s = 1 anonly if (R n v 1 , R n v 2 )

∈

E (G). We get a more applicable dimE G (n1, · · ·, nm ) on K 3-free graphs G by Theorem 4 .1.1.

Corollary 4.1.1 If G is K 3-free, then

dimE G (n1, · · ·, nm ) =v∈V (G)

nv −(u,v )∈E (G)dim(R

Particularly, if G = v1v2 vm a circuit for an integer m



· · · ≥dimE G (n1, · · ·, n m ) =m

i=1nvi −

m

i=1dim(R n v i

where each index is modulo m.

Now we determine the maximum and minimum dimenEuclidean spaces of R n 1 , R n 2 ,

· · ·, R n m with an underlying s

Theorem 4.1.2 Let E G (nv1 , · · ·, n vm ) be a combinatorial EuR n v 2 , · · ·, R n v m with an underlying graph G, V (G) = {v1

maximum dimension dimmax E G (nv1 , · · ·, nvm ) of E G (nv1 , · ·dimmax E G (nv1 , · · ·, nvm ) = 1 −m +

v∈V (Gwith conditions dim(R n u ∩R n v ) = 1 for ∀(u, v)∈E (G).

Proof Let X v1 , X v2 , · · ·, X vm consist of these orthogonaR n v 2 , · · ·, R n v m , respectively. Notice that

|X vi X vj | = |X vi |+ |X vj | − |X vi Xfor 1 ≤ i = j ≤ m by Theorem 1 5 1 in the case of n = 2

158 Chap.4 C

with {v1, v2}in G. Then by

|X v1 X v2 X v3 | = |X v1 X v2 |+ |X v3 | − |(X v1 X v2

we know that |X v1∪X v2∪X v3 | attains its maximum value only if its maximum and |(X v1 ∪X v2 ) ∩X v3 | = 1 for (X v1 ∪X v2 ) ∩X

|X v1 X v3

|= 1 or

|X v2 X v3

|= 1, or both. In the later cas

|X X X | = 1 Therefore the maximum value of |X ∪X



| ∩ | | ∩ ||X v1 ∩X v2 ∩X v3 | = 1. Therefore, the maximum value of |X v1∪X

|X v1 |+ |X v2 |+ |X v3 | −2.

Generally, we assume the maximum value of |X v1∪X v2∪ ·

|X v1 |+ |X v2 |+ · · ·+ |X vk | −k + 1

for an integer k ≤ m with conditions |X vi ∩X vj | = 1 hold if (1 ≤ i = j ≤k. By the connectedness of G, without loss of genervertex vk+1 adjacent with {v1, v2, · · ·, vk}in G and nd out the m

|X v1

∪

X v2

∪ · · ·∪X vk

∪

X vk +1

|. In fact, since

|X v1∪X v2∪ · · ·∪X vk ∪X vk +1 | = |X v1∪X v2∪ · · ·∪X vk

− |(X v1∪X v2∪ · · ·∪X v

we know that

|X v1

∪

X v2

∪ · · ·∪X vk

∪

X vk +1

|attains its maxim

|X v1∪X v2∪ · · ·∪X vk | attains its maximum and |(X v1∪X v2∪ · ·for (X ∪X ∪ ∪X )∩X = ∅Whence |X ∩X | = 1 if


Determining the minimum value dim min E G (n1, · · ·, n m )difficult problem in general case. But we can still get it for s

Theorem 4.1.3 Let E G (nv1 , n v2 , · · ·, nvm ) be a combinatoR n v 1 , R n v 2 , · · ·, R n v m with an underlying graph G, V (G)

{v1, v2, · · ·, vl}an independent vertex set in G. Then

dimmin E G (nv1 nvm ) ≥l

nvi



dim E (n , · · ·, n ) ≥ i=1 n

and with the equality hold if G is a complete bipartite graphsets V 1 = {v1, v2, · · ·, vl}, V 2 = {vl+1 , vl+2 , · · ·, vm}and

l

i=1

nvi

≥

m

i= l+1

nvi .

Proof Similarly, we use X v1 , X v2 , · · ·, X vm to denote thtions in R n v 1 , R n v 2 , · · ·, R n v m , respectively. By denition, w

X vi X vj =

∅

, 1

≤i = j

≤l

for (vi , v j )∈E (G). Whence, we get that

|m

i=1

X vi | ≥ |l

i=1

X vi | =l

i=1

nvi .

By the assumption,

l m

160 Chap.4 C

such thatl

k=1 |Y i(vk)| = |X vi | for any integer i, l + 1 ≤ i ≤m, w

an empty set for integers i, 1 ≤ i ≤ l. Whence, we can choose

X vi =l

k=1Y i(vk)

to replace each X vi for any integer i, 1 ≤ i ≤m. Notice that the iof X v1 , X v2 , · · ·, X vl , X vl +1

, · · ·, X vm is still the complete bipartite but

m l l



|m

i=1X vi | = |

l

i=1X vi | =

l

i=1n i .


dimmin E G (nv1 ,

· · ·, n vm ) =

l

i=1

nvi

in the case of complete bipartite graph K (V 1, V 2) with partite sets VV 2 = {vl+1 , vl+2 , · · ·, vm}and

l

i=1

nvi ≥m

i= l+1

nvi .

Although the lower bound of dim E G (nv1 , · · ·, n vm ) in Theorbut sometimes this bound is not better if G is given, for examgraph K m shown in the next results. Consider a complete systemset with less than 2 r elements. We know the next conclusion.

Theorem 4.1.4 For any integer r ≥ 2, let E K m (r ) be a combif R r R r d h i i 0 1


r + s −1

r< m

≤r + s

rand 0 ≤s ≤r −1, we know that two r -subsets of an (r + s)-seintersection. So we can determine these m r -subsets X 1, Xcomplete system of r -subsets in an ( r + s)-set, and these m r -can not be chosen in an ( r + s −1)-set. Therefore, we nd t

m



|i=1

X i| = r + s,

i.e., if 0≤s ≤r −1, then

dimmin E K m (r ) = r + s

Because of our living world is the space R 3, so the combparticularly interesting in physics. We completely determine itin the case of K m following.

Theorem 4.1.5 Let E K m (3) be a combinatorial Euclidean spa

dimmin E K m (3) =⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

3, if m = 1 ,4, if 2≤m5, if 5≤m

2 + √m , if m ≥1

162 Chap.4 C

We only consider the case of m ≥ 11. Let X = {u,v,w}Notice that any 3-set will intersect X with 1 or 2 elements. Our diinto three cases.

Case 1 There exist 3-sets X 1, X 2, X 3 such that X 1 ∩X = {u, v}and X 3 ∩X = {v, w}such as those shown in Fig.4.1.1, where eaca 3-set.



wu

v

Fig. 4.1.1

Notice that there are no 3-sets X such that |X ∩X | = 1 in thiwe can easily nd two 3-sets with an empty intersection, a contradsuch 3-sets, we know that there are at most 3( v−3)+1 3-sets withgraph being K m . Thereafter, we know that

m ≤3(l −3) + 1 , i.e., l ≥m −1

3+ 3 .

Case 2 There are 3-sets X 1, X 2 but no 3-set X 3 such that XX 2

∩X =

{u, w

}and X 3

∩X =

{v, w

}such as those shown in Fig

triangle denotes a 3-set.


2(l

−1) +

l −3

2+ 1

3-sets with their intersection graph still being K m . Whence,

m ≤2(l −1) +l −3

2+ 1 , i.e., l ≥

3 +

Case 3 There are a 3-set X 1 but no 3-sets X 2, X 3 such



1 2, 3

X 2 ∩X = {u, w}and X 3 ∩X = {v, w}such as those shown triangle denotes a 3-set.

u vw

Fig. 4.1.3

Enumerating 3-sets in this case, we know that there are

l −2 + 2l −2

2

such 3-sets with their intersection graph still being K m . The

m ≤ l −2 + 2l

−2

2 , i.e., l ≥2 +

164 Chap.4 C


dimmin E K m (3) = 2 + √m

if m ≥11. This completes the proof.For general combinatorial spaces E K m (nv1 , · · ·, nvm ) of R n v 1

we get their minimum dimension if nvm is large enough.

Theorem 4.1.6 Let E K m be a combinatorial Euclidean space ofm +1



R n v m , nv1 ≥nv2 ≥ · · · ≥nvm ≥ log2( m +12n v 1 − n v 2 − 1 ) +1 and V (K m )

Then

dimmin E K m (nv1 , · · ·, nvm ) = nv1 + log2(m + 1

2n v 1 − n v 2 − 1

Proof Let X v1 , X v2 , · · ·, X vm be sets consist of these orthogonR n v 1 , R n v 2 , · · ·, R n v m , respectively and

2s− 1 <m

2k+1 −1+ 1 ≤2s

for an integer s, where k = nv1

−nv2 . Then we nd that

log2(m + 1

2n v 1 − n v 2 − 1 ) = s.

We construct a family {Y v1 , Y v2 , · · ·, Y vm }with none being a

|Y vi | = |X vi | for 1 ≤ i ≤m and its intersection graph is still K m ,

|Y v1 Y v2 · · · Y vm | = nv1 + s.


Choose g elements xi1 , x i2 , · · ·, x ig ∈X v1 and h ≥1 elemU . We construct a nite set

X g.h = {xi1 , x i2 , · · ·, x ig , u j 1 , u j 2 , · · ·, u j

with a cardinal g + h. Let g + h = |X v1 |, |X v2 |, · · ·, |X vm |sequently nd such sets Y v1 , Y v2 , · · ·, Y vm . Notice that therea subset of another in the family

{Y v1 , Y v2 ,

· · ·, Y vm

}. So t

ements in each Y vi , 1 ≤ i ≤ m at least such that one is



{ }≤ ≤{xn v 2

, xn v 2 +1 , · · ·, xn v 1 }. Now since nvm ≥ log2( m +12n v 1 − n v 2 − 1 )

k+1

i=1

s

j =1

k + 1i

s j

= (2 k+1 −1)(2s −different sets Y v1 , Y v2 , · · ·, Y vm altogether with |X v1 | = |Y v1 |, ·of them is a subset of another and their intersection graph is s

X v1 , {u1, x1, · · ·, xn v 2 − 1},

{u1, xn v 2 − n v 3 +2 , · · ·, xn v 2 },

· · · · · · · · · · · · · · · · · ·,

{u1, xn v k − 1 − n v k +2 , · · ·, xn v k }are such sets with only one element u1 in U . See also in Feasily to know that

|Y v1 Y v2 · · · Y vm | = nv1 + s = nv1 + log2( 2

166 Chap.4 C

different sets in {Y v1 , Y v2 , · · ·, Y vm }with none being a subset of anothat there must exists integers i,j, 1

≤i = j

≤m with Y vi

⊂

Y vTherefore, we get the minimum dimension dim min E K m of E K m to

dimmin E K m (nv1 , · · ·, n vm ) = nv1 + log2(m + 1

2n v 1 − n v 2 −

4.1.2 Combinatorial Fan-Space. A combinatorial fan-spacethe combinatorial Euclidean space E K (n1 nm ) of R n 1 R n 2



the combinatorial Euclidean space K m (n1, · · ·, nm ) of R , R ,for any integers i,j, 1 ≤ i = j ≤m,

R n i R n j =m

k=1

R n k ,

which is applied for generalizing n-manifolds to combinatorial msection. The dimensional number of R (n1, · · ·, nm ) is determinedenition following.

Theorem 4.1.7 Let R (n1, · · ·, nm ) be a fan-space. Then

dimR (n1, · · ·, nm ) = m +m

i=1

(n i −m),

where

m = dim(m

k=1

R n k ).


(A), (B) =i,j

a ij bij .

Then we know

Theorem 4.1.8 Let (A), (B), (C ) be m ×n matrixes and α(1) A, B = B, A ;(2) A + B, C = A, C + B, C ;(3) αA,B = α B, A ;



(4) A, A ≥0 with equality hold if and only if (A) =

Proof (1)-(3) can be gotten immediately by denition. Nthat

A, A =i,j

a2ij ≥0

and with equality hold if and only if a ij = 0 for any integersn, namely, ( A) = Om × n .

By Theorem 4.1.8, all matrixes of real entries under th

Euclidean space. We also generalize some well-known resultspace. The rst, Theorem 3 .2.1(i) is generalized to the next

Theorem 4.1.9 Let (A), (B) be m ×n matrixes. Then

(A), (B) 2

≤(A), (A) (B), (B)

and with equality hold only if (A) = λ(B), where λ is a real

168 Chap.4 C

Therefore, we nd that

Δ = ( −2 (A), (B) )2 −4 (A), (A) (B), (B) <

namely,

(A), (B) 2 < (A), (A) (B), (B) .



Corollary 4.1.2 For given real numbers a ij , bij , 1 ≤ i ≤m, 1 ≤ j

(i,j

a ij bij )2 ≤(i,j

a2ij )(

i,j

b2ij ).

Now let O be the original point of R (n1, · · ·, nm ). Then [O

∀ p,q∈R (n1, · · ·, nm ), we also call −→Op the vector correspondent tolar to that of Euclidean spaces, Then −→ pq = −→Oq−−→Op. Theorem 4.1troduce an angle between two vectors −→ pq and −→uv for points p,q,u,v

Let p,q,u,v∈R (n1, · · ·, nm ). Then the angle θ between ve

determined by

cos θ =[ p]−[q], [u]−[v]

[ p]−[q], [ p]−[q] [u]−[v], [u]−[v]under the condition that 0 ≤θ ≤π.

Corollary 4.1.3 The conception of angle between two vectors is

Proof Notice that



170 Chap.4 C

hold for an integer m, 1 ≤m ≤n. Then there is a combinatorial fansuch that

R n∼= R (n1, n2, · · ·, nm ).

Proof Not loss of generality, assume the normal basis of R n i

2 = (0 , 1, 0 · · ·, 0), · · ·, n = (0 , · · ·, 0, 1). Then its coordinate

(x1, x2, · · ·, xn ). Since



n −m =m

i=1

(n i −m),

choose

R 1 = 1, 2, · · ·, m ,

m +1 , · · ·, n 1 ;

R 2 = 1, 2, · · ·, m , n 1 +1 , n 1 +2 , · · ·, n 2 ;

R 3 = 1, 2, · · ·, m , n 2 +1 , n 2 +2 , · · ·, n 3 ;

· · · · · · · · · · · · · · · · · · · · · · · · · ·;

R m = 1, 2,

· · ·, m , n m − 1 +1 , n m − 1 +2 ,

· · ·, n m

Calculation shows that dim R i = n i and dim(m

R i) = m Wh

Sec.4.2 Combinatorial Manifolds

Theorem 4.1.12 Let GE be an edge labeled graph on a conneing θE : E (G) →[1, l]. If nv, v∈V (G) are given integers wit

then there are sets X v, v∈V (G) such that |X v| = nv and |Xv∈V (G), u∈N G (v).

Proof For (v, u)∈E (G), construct a nite set

(v, u ) =

{(v, u)1, (v, u )2,

· · ·, (v, u )θE (v,u

Now we dene



X v = (u∈N G (v)

(v, u )) {x1, x2, · · ·, xς

for ∀v ∈V (G), where ς = nv − u∈N G (v)θE (v, u). Then w

X v, v∈V (G) satisfy |X v| = nv, |X v∩X u | = θE (v, u) for∀v∈This completes the proof.

As a special case, choosing the labeling 1 on each edge we get the result of Erd os et al. again.

Corollary 4.1.4 For any graph G, there exist sets X v, v

∈

V (graph G, i.e., the minimum number of elements in X v, v∈Vto ε(G).

Calculation shows that

| v∈V (G)

X v

|=

v∈V (G)

nv

−1

2 (v,u )∈E (G)

θE (v

in the construction of Theorem 4 1 12 we get a decompositio

172 Chap.4 C

§4.2 COMBINATORIAL MANIFOLDS

4.2.1 Combinatorial Manifold. For a given integer sequence n1 with 0 < n 1 < n 2 < · · ·< n m , a combinatorial manifold M is such that for any point p∈M , there is a local chart ( U p,ϕ p) of p, iborhood U p of p in M and a homoeomorphism ϕ p : U p →R (n1( p),a combinatorial fan-space with

{n1( p), n2( p), · · ·, n s( p)( p)}⊆ {n1, n 2, · · ·, nm},



and

p∈M {n1( p), n 2( p), · · ·, n s ( p)( p)}= {n1, n2, · · ·, nm

denoted by M (n1, n 2,

· · ·, nm ) or M on the context, and

A= {(U p,ϕ p)| p∈M (n1, n 2, · · ·, n m ))}an atlas on M (n1, n 2, · · ·, nm ). The maximum value of s( p) a

s( p) = dim(

s( p)

i=1R n i ( p)) are called the dimension and the intersectio

M (n1, n2, · · ·, nm ) at the point p, respectively.A combinatorial manifold M is nite if it is just combined b

with an underlying combinatorial structure G without one manithe union of others. Certainly, a nitely combinatorial manifold isnatorial manifold.

Two examples of such combinatorial manifolds with different are shown in Fig.4.2.1, in where, (a) represents a combination o


By denition, combinatorial manifolds are a generalizacombinatorial speculation. However, a compact n-manifold

is itself a combinatorial Euclidean space E G (n, · · ·, n

m

) of Eu

an underlying structure G shown in the next result.

Theorem 4.2.1 A compact n-manifold M n without boundaa combinatorial Euclidean space E G (n, · · ·, n

m

) of spaces R n

on M n .



Proof Let

A= {(U p,ϕ p)| ϕ p : U p →R n ,∀ p∈M n

be an atlas of M n . By denition, M n is compact. Whence,

with only nite charts, i.e., there is an integer 1 ≤m ≤+ ∞A[m ] = {(U i ,ϕi)|1 ≤ i ≤m}

is a nite atlas on M n . Therefore, we can dene an underlyinture G by

V (G) = {U i , 1 ≤ i ≤m},

E (G) = {(U i , U j )|U i U j = ∅, 1 ≤ i = j ≤Then we get a combinatorial manifold M (n) underlying the

Now we can also dene a combinatorial Euclidean space

R n by




1 3 2

1 3 2

1

1

1

1

0

0

2

2

1 2

2

0

0

0

00

0

0

(a) (b)

(c) (d)



( ) ( )

Fig. 4.2.2

Theorem 4.2.2 Let Gd[M (n1, n 2, · · ·, nm )] be a labelled gr

natorial manifold M (n1, n2, · · ·, nm ). Then (1) Gd[M (n1, n 2, · · ·, n m )] is connected only if d ≤n1.(2) there exists an integer d, d ≤ n1 such that Gd[M (

nected.

Proof By denition, there is an edge ( M n i , M n j ) in G

1 ≤ i, j ≤ m if and only if there is a d-dimensional path Ppoints p∈M n i and q∈M n j . Notice that

(P d( p,q) \ M n i )⊆M n j and (P d( p,q) \ M n j )

Whence,

d ≤min{n i n j}

176 Chap.4 C

by (4.2.1). However, by denition we know that

p∈M {n1( p), n2( p), · · ·, n s( p)( p)}= {n1, n 2, · · ·, n m}. (


d

≤min(

p∈M {n1( p), n 2( p),

· · ·, n s( p)( p)

}) = min

{n1, n 2,

· ·b bi i (4 2 2) i h (4 2 3) N i h i l b l d i h 0



by combining (4.2.2) with (4.2.3). Notice that points labeled with 0connected by a path. We get the conclusion (1).

For the conclusion (2), notice that any nitely combinatoriaways pathwise 1-connected by denition. Accordingly, G1[M (n1, nnected. Thereby, there at least one integer, for instance d = 1 enab

· · ·, n m )] to be connected. This completes the proof.According to Theorem 4 .2.2, we get immediately two corolla

Corollary 4.2.1 For a given nitely combinatorial manifold M , alGd[M ] are isomorphic if d

≤n1, denoted by GL [M ].

Corollary 4.2.2 If there are k 1-manifolds intersect at one poicombinatorial manifold M , then there is an induced subgraph K k

Now we dene an edge set E d(M ) in GL [M ] by

E d

(M ) = E (Gd

[M ]) \ E (Gd+1

[M ]).Then we get a graphical recursion equation for graphs of a nite


Now let H(n1, n2, · · ·, n m ) denote all nitely combinatori

· · ·, nm ) and G[0, n m ] all vertex-edge labeled graphs GL with

{0, 1, · · ·, nm}with conditions following hold.

(1)Each induced subgraph by vertices labeled with 1 in Ggraphs and vertices labeled with 0 can only be adjacent to ve

(2)For each edge e = ( u, v)∈E (G), τ 2(e) ≤min{τ 1(u),

Then we know a relation between sets

H(n1, n 2,

· · ·, nm

following.



Theorem 4.2.4 Let 1 ≤ n1 < n 2 < · · ·< n m , m ≥ 1quence. Then every nitely combinatorial manifold M ∈ nes a vertex-edge labeled graph G([0, nm ]) ∈ G[0, nm ]. Coedge labeled graph G([0, nm ])

∈ G[0, n m ] denes a nitely c

M ∈ H(n1, n 2, · · ·, n m ) with a 1 −1 mapping θ : G([0, n m

is a θ(u)-manifold in M , τ 1(u) = dim θ(u) and τ 2(v, w) =

∀u∈V (G([0, nm ])) and ∀(v, w)∈E (G([0, nm ])).

Proof By denition, for ∀M ∈H(n1, n 2, · · ·, nm ) there

graph G([0, nm ])∈ G([0, nm ]) and a 1 −1 mapping θ : M θ(u) is a θ(u)-manifold in M . For completing the proof, wnitely combinatorial manifold M ∈ H(n1, n2, · · ·, nm ) forwith τ 1(u) = dim θ(u) and τ 2(v, w) = dim( θ(v) θ(w)) for ∀∀(v, w)∈E (G([0, nm ])). The construction is carried out by p

STEP 1 . Choose |G([0, n m ])| − |V 0| manifolds correspondena dimensional n i if τ 1(u) = n i , where V 0 = {u|u ∈V (G([0

178 Chap.4 C

(N G([0,n m ])(ul+1 ) V ≥ 1)

\Δ l =

{v1

l+1 , v2l+1 ,

· · ·, vs (u l

l+1

with τ 1(v1l+1 ) = n l+1 ,1, τ 1(v2

l+1 ) = n l+1 ,2, · · ·,τ 1(vs (u l +1 )l+1 ) = n l+1 ,s (u

manifold correspondent to the vertex ul+1 with an intersection dimewith the manifold correspondent to the vertex vi

l+1 , 1 ≤ i ≤ s(vertex set Δ l+1 = Δ l {ul+1 }.

STEP 4 . Repeat steps 2 and 3 until a vertex set Δ t = V ≥ 1 has This construction is ended if there are no vertices w∈V (G) wi



V ≥ 1 = V (G). Otherwise, go to the next step.

STEP 5 . For ∀w∈V (G([0, nm ])) \ V ≥ 1, assume N G([0,n m ])(w) =Let all these manifolds correspondent to vertices w1, w2, · · ·, we

point simultaneously and dene a vertex set Δ∗

t+1 = Δ t {w}.STEP 6 . Repeat STEP 5 for vertices in V (G([0, nm ])) \ V ≥ 1. Tnally ended until a vertex set Δ ∗

t+ h = V (G[n1, n2, · · ·, nm ]) has b

A nitely combinatorial manifold M correspondent to G([0, nΔ∗

t+ h has been constructed. By this construction, it is easily ve

H(n1, n 2, · · ·, nm ) with τ 1(u) = dim θ(u) and τ 2(v, w) = dim( θ(v)V (G([0, nm ])) and ∀(v, w)∈E (G([0, nm ])). This completes the pr

4.2.2 Combinatorial Submanifold. A subset S of a combM is called a combinatorial submanifold if it is itself a combinatoGL [S ]

≺

GL [M ]. For nding some simple criterions of combinatorwe only consider the case of F : M →N mapping each manifol


4

2 3 3

4

3 4

2 2

3

2

1 1 2

42 1

3 42 2

(a) (b)

Fig. 4.2.3

f f



For characterizing combinatorial in-submanifolds of a coM , we introduce the conceptions of feasible vertex-edge labequotient graph in the following.

Denition 4.2.2 Let M be a nitely combinatorial maning graph GL [M ]. For ∀M ∈V (GL [M ]) and U L ⊂N GL

τ 2(M, M i) ≤ τ 2|GL [M ](M, M i ) for ∀M i ∈U L , let J (M i) =τ 2(M, M i), M i ⊂M i}and denotes all these distinct represenU L by T . Dene the index oM (M : U L ) of M relative to U

oM (M : U L ) = minJ ∈T {dim(

M ∈J

(M M )

A vertex-edge labeled subgraph ΓL of GL [M ] is feasible

τ 1|Γ(u) ≥oM (u : N ΓL (u)).

Denoted by ΓL≺o GL [M ] a feasibly vertex-edge labeled subg

180 Chap.4 C

F 11 (M 1)⊂N 1, F 11 (M 2)⊂N 2 and F 11 (M 1)

and labeling each vertex N with dimM if F 11 (M )

⊂

N and each edim (M 1 ∩M 2) if F 1(M 1)⊂N 1, F 11 (M 2)⊂N 2 and F 11 (M 1) ∩F 11 (M

Then, we know the following criterion on combinatorial subm

Theorem 4.2.5 Let M and N be nitely combinatorial manifolcombinatorial in-submanifold of N if and only if there exists an i

such that



GL [M ]/F 11 ≺o N.

Proof If M is a combinatorial in-submanifold of N , by de

that there is an injection F : M →N such that F (M ) ∈V V (GL [M ]) and there are no two different N 1, N 2 ∈

L with FF 11 (M ) ∩N 2 = ∅. Choose F 11 = F . Since F is locally 1−1 weM 2) = F (M 1) ∩F (M 2), i.e., F (M 1, M 2) ∈E (G[N ]) or V (G[N ]E (GL [M ]). Whence, GL [M ]/F 1

1 ≺GL [N ]. Notice that GL [M ] is c

M . Whence, it is a feasible vertex-edge labeled subgraph of GL

Therefore, GL [M ]/F 11 ≺o GL [N ].

Now if there exists an injection F 11 on M , let ΓL≺o GL [

the graph GL [N ] \ ΓL , where GL [N ] \ ΓL denotes the vertex-edgeinduced by edges in GL [N ]\ΓL with non-zero labels in G[N ]. We M ∗of N by


Corollary 4.2.3 For two nitely combinatorial manifolds Mtorial monotonic submanifold of N if and only if GL [M ]≺o

Proof Notice that F 11 ≡ 111 in the monotonic case.

GL [M ]/ 111 = GL [M ]. Thereafter, by Theorem 4 .2.9, we kno

torial monotonic submanifold of N if and only if GL [M ]≺o

4.2.3 Combinatorial Equivalence. Two nitely combina

n2, · · ·, nm ), M 2(k1, k2, · · ·, kl) are called equivalent if thesegraphs



GL [M 1(n1, n2, · · ·, nm )]∼= GL [M 2(k1, k2, · ·Notice that if M 1(n1, n 2, · · ·, n m ), M 2(k1, k2, · · ·, kl) are

get that {n1, n2, · · ·, nm}= {k1, k2, · · ·, kl}and GL [M 1]∼=idea enables us classifying nitely combinatorial manifolds inthe action of automorphism groups of these correspondent gr

Denition 4.2.4 A labeled connected graph GL [M (n1, n2, ·ally unique if all of its correspondent nitely combinatorial ma

are equivalent.

Denition 4.2.5 A labeled graph G[n1, n 2, · · ·, n m ] is calleautomorphism group Aut G is transitive on {C (n i ), 1 ≤ i ≤mall these vertices with label n i .

We nd a characteristic for combinatorially unique graph

Theorem 4 2 6 A labeled connected graph G[n n

182 Chap.4 C

C θ(n i) = C (n j ) for∀i,j, 1 ≤ i, j ≤m.

On the other hand, if G[n1, n2, · · ·, nm ] is class-transitive,

i,j, 1 ≤ i, j ≤ m, there is an automorphism τ ∈Aut G such thatWhence, for any re-labeled graph G [n1, n2, · · ·, nm ], we nd that

G[n1, n2, · · ·, n m ]∼= G [n1, n2, · · ·, nm ],

which implies that these nitely combinatorial manifolds correspon

· · ·, n m ] and G [n1, n 2,

· · ·, nm ] are combinatorially equivalent, i.e.,

is combinatorially unique.

Now assume that for parameters t i1 t i2 t is we have kno



Now assume that for parameters t i1, t i2, · · ·, t is i , we have kno

C M n i [xi1, x i2, · · ·] =t i 1 ,t i 2 ,··· ,t is

n i(t i1, t i2, · · ·, t is )xt i 1i1 xt i 2

i2

for n i -manifolds, where n i(t i1, t i2, · · ·, t is ) denotes the number of nn i-manifolds with parameters t i1, t i2, · · ·, t is . For instance the enupact 2-manifolds with parameter genera is

C M [x](2) = 1 + p≥ 1

2x p.

Consider the action of Aut G[n1, n2, · · ·, nm ] on G[n1, n2, · · ·, nof orbits of the automorphism group Aut G[n1, n2, · · ·, nm ] action m}is π0, then we can only get π0! non-equivalent combinatorialspondent to the labeled graph G[n1, n 2, · · ·, n m ] similar to Theotion shows that there are l! orbits action by its automorphism gro

(s1 + s2 + · · ·+ s l)-partite graph K (ks 11 , k

s 22 , · · ·, k

s ll ), where k

s ii d

are s i partite sets of order ki in this graph for any integer i 1 ≤


i ≤ m}, then the enufunction of combinatorial manifolds Mspondent to a labeled graph G[n1, n 2, · · ·, nm ] is

C M (x) = π0!m

i=1

C M n i [xi1, x i2, · · ·],particularly, if G[n1, n 2, · · ·, n m ] = K (ks1

1 , ks 22 , · · ·, ks m

m ) such tite sets labeled with n i is s i for any integer i, 1 ≤ i ≤ mcorrespondent to K (ks1

1, ks 2

2,

· · ·, ks m

m) is

C (x) = m!m

CM n i [xi1 x i2 ]



C M (x) m!i=1

C M n i [xi1, x i2, · · ·]and the enufunction correspondent to a complete graph K m

C M (x) =m

i=1

C M n i [xi1, x i2, · · ·].

Proof Notice that the number of non-equivalent nitely cocorrespondent to G[n1, n 2, · · ·, nm ] is

π0

m

i=1n i(t i1, t i2, · · ·, t is )

for parameters t i1, t i2, · · ·, t is , 1 ≤ i ≤ m by the product priWhence, the enufunction of combinatorial manifolds M (n1

dent to a labeled graph G[n1, n2, · · ·, nm ] is

m m

184 Chap.4 C

g : M (n1, n2, · · ·, nm ) →M (k1, k2, · · ·, kl)

such that gf identity: M (k1, k2, · · ·, kl) →M (k1, k2, · · ·, kl) aM (n1, n2, · · ·, nm ) →M (n1, n 2, · · ·, nm ).

For equivalent homotopically combinatorial manifolds, we knresult.

Theorem 4.2.8 Let M (n1, n2,

· · ·, nm ) and M (k1, k2,

· · ·, kl) be

rial manifolds with an equivalence : GL [M (n1, n2, · · ·, nm )] →GIf for ∀M 1, M 2 ∈V (GL [M (n1, n2, · · ·, nm )]), M i is homotopic t



∀ ∈motopic mappings f M i : M i → (M i), gM i : (M i) →M i such f M j |M i M j , gM i |M i M j = gM j |M i M j providing (M i , M j )∈E (GL [ for 1

≤i, j

≤m, then M (n1, n2,

· · ·, nm ) is homotopic to M (k1, k

Proof By the Gluing Lemma, there are continuous mappings

f : M (n1, n 2, · · ·, nm ) →M (k1, k2, · · ·, kl)

and

g : M (k1, k

2,

· · ·, k

l)

→M (n

1, n

2,

· · ·, n

m)

such that

f |M = f M and g| (M ) = g (M )

for∀M ∈V (GL [M (n1, n2, · · ·, nm )]). Thereby, we also get that

gf identity : M (k1, k2, · · ·, kl) →M (k1, k2, · · ·,d


4.2.5 Euler-Poincare Characteristic. It is well-known

χ (M ) =∞

i=0(−1)iα i

with α i the number of i-dimensional cells in a CW -complexEuler-Poincare characteristic of this complex. In this subsectiPoincare characteristic for nitely combinatorial manifolds. Fora clique sequence

{Cl(i)

}i≥ 1 in the graph GL [M ] by the foll

STEP 1 . Let Cl(GL [M ]) = l0. Construct



Cl(l0) = {K l01 , K l0

2 , · · ·, K i0 p |K l0

i GL [M ] and K

or a vertex∈V(GL[M]) for i = j, 1 ≤ i

STEP 2 . Let G1 =K l∈Cl (l)

K l and Cl(GL [M ] \ G1) = l1. Co

Cl(l1) = {K l11 , K l1

2 , · · ·, K i1q |K l1

i GL [M ] and K

or a vertex∈V(GL[M]) for i = j, 1 ≤STEP 3 . Assume we have constructed Cl(lk− 1) for an inte

K l k − 1∈Cl (l)K lk − 1 and Cl(GL [M ] \ (G1∪ · · ·∪Gk )) = lk . We c

Cl(lk) = {K lk1 , K lk

2 , · · ·, K lkr |K lk

i GL [M ] and K

or a vertex∈V(GL[M]) for i = j, 1 ≤ i

186 Chap.4 C

Proof Denoted the numbers of all these i-dimensional cells manifold M or in a manifold M by α i and α i (M ). If GL [M ]

complete graph K k with V (GL [M ]) = {M 1, M 2, · · ·, M k}, k ≥ 2inclusion-exclusion principe and the denition of Euler-Poincare get that

χ (M ) =∞

i=0

(−1)i α i

=∞

i=0

(−1)i

M ∈V(K k ) 1≤ j ≤ ≤ k

(−1)s+1 α i(M i1 ·



i=0 M i j ∈V (K k ),1≤ j ≤ s≤ k

=M i j ∈V (K k ),1≤ j ≤ s≤ k

(−1)s+1∞

i=0

(−1)iα i(M i1 ·=

M i j ∈V (K k ),1≤ j ≤ s≤ k

(−1)s+1 χ (M i1 · · · M is )

for instance, χ (M ) = χ (M 1)+ χ (M 2)−χ (M 1∩M 2) if GL [M ] = K 2

{M 1, M 2}. By the denition of clique sequence of GL [M ], we na

χ (M ) =K k∈Cl (k),k≥ 2 M i j ∈V (K k ),1≤ j ≤ s≤ k

(−1)i+1 χ (M i1 · ·

If GL [M ] is just one of some special graphs, we can get interesby Theorem 4 .2.14.

Corollary 4.2.4 Let M be a nitely combinatorial manifold. If

Sec.4.3 Fundamental Groups of Combinatorial Manifolds

Proof Notice that GL [M ] is K 3-free, we get that

χ (M ) =(M 1 ,M 2 )∈E (GL [M ])

(χ (M 1) + χ (M 2) −χ (M 1 M

=(M 1 ,M 2 )∈E (GL [M ])

(χ (M 1) + χ (M 2)) −(M 1 ,M 2 )∈E (

=M ∈V (GL [M ])

χ 2(M )

−(M 1 ,M 2 )∈E (GL [M ])

χ (M 1



Since the Euler-Poincare characteristic of a manifold M iwe get the following consequence.

Corollary 4.2.5 Let M be a nitely combinatorial manifonumber for any intersection of k manifolds with k ≥2. The

χ (M ) =M ∈V (GL [M ])

χ (M ).

§4.3 FUNDAMENTAL GROUPS OF

COMBINATORIAL MANIFOLDS

4.3.1 Retraction. Let ϕ : X

→Y be a continuous ma

spaces X to Y and a, b : I →X be paths in X . It is readily

188 Chap.4 C

Such a ϕ∗ is called a homomorphism induced by ϕ, particularinduced by ϕ if ϕ is an isomorphism.

Denition 4.3.1 A subset R of a topological space S is called a rexists a continuous mapping o : S →R, called a retraction such

∀a∈R.

Now let o : S →R be a retraction and i : R →S a inclusion point x

∈

R, we consider the induced homomorphism

o∗ : π(S, x ) →π(R, x ), i∗ : π(R, x ) →π(S, x )

Notice that oi =identity mapping by denition which implies that



Notice that oi =identity mapping by denition, which implies thatmapping of the group π(R, x 0) by properties ( iv) and (v) previou

Denition 4.3.2 A subset R of a topological space S is called a dof S if there exists a retraction o : S →R and a homotopy f : S

f (x, 0) = x, f (x, 1) = o(x) for ∀x∈S,

f (a, t ) = a fo r

∀

a

∈

R, t

∈

I.

Theorem 4.3.1 If R is a deformation retract of a topological sinclusion mapping i : R →S induces an isomorphism of π(R, x 0

∀x0∈R, i.e., π(R, x 0)∼= π(S, x 0)

Proof As we have just mentioned, o∗i∗ is the identity mappiio : X →X is an identity mapping with io(x0) = x0. Whence,


4.3.2 Fundamental d-Group. Let a nitely combinator

· · ·, nm ) be d-arcwise connected for some integers 1 ≤d ≤n

tal group, we consider fundamental d-groups of nitely comthis subsection.

Denition 4.3.4 Let M (n1, n2, · · ·, nm ) be a nitely combarcwise connectedness for an integer d, 1 ≤d ≤n1 and ∀x0

fundamental d-group at the point x0, denoted by πd(M (n1, n2

to be a group generated by all homotopic classes of closed d-

If d = 1 and M (n1, n2, · · ·, nm ) is just a manifold M , w



πd(M (n1, n2, · · ·, nm ), x) = π1(M, x ).

Whence, fundamental d-groups are a generalization of fundasical topology.

A combinatorial Euclidean space E G ( d,d, · · ·, dm

) of R d

torial structure G, |G| = m is called a d-dimensional graph ,

(1) M d[G]

\V (M d[G]) is a disjoint union of a nite n

e1, e2, · · ·, em , each of which is homeomorphic to an open ba(2) the boundary ei −ei of ei consists of one or two ver

(ei , ei) is homeomorphic to the pair ( Bd, S d− 1),

The next result is gotten by denition.

Theorem 4.3.2 πd

(M d

[G], x0)∼= π1(G, x 0), x0∈G.

F d t i i g th d f d t l g f bi t i

190 Chap.4 C

Theorem 4.3.3 Let M (n1, n2, · · ·, nm ) be a nitely combinatorialying a combinatorial structure G, M d[G]≺M (n1, n2, · · ·, nm ) su

M (n1, n 2, · · ·, nm ) \ M d[G] =k

i=2

li

j =1

B i j ,

x0∈M d[G]. Then

πd(M (n1, n2, · · ·, nm ), x0)∼= π1(G, x 0)β 2j α2j β − 1

2j |1 ≤ j ≤ l2

where α2 is the closed path of B2 and β2 a path in X with an in



where α2j is the closed path of B2j and β 2j a path in X with an interminal point on α2j .

Proof For any s-ball Bsj , 1

≤j

≤ls , choose one point

ne U = M (n1, n 2, · · ·, n m ) \ {us0j }and V = Bsj . Then U , V M (n1, n2, · · ·, nm ) = U ∪V . Notice that U, V , V ∩V = Bsj {us0j

nected and V simply connected. Applying Corollary 3 .1.2 and Tget that

πd(M (n1, n2,

· · ·, nm ), x0)

∼

= π (G,x 0 )

π 1 (U ∩V )N = π (G,x 0 )

i1∗(π1 (B sj {u s 0

Since

π1(Bsj {us0j }) =Z, if s = 2 ,

{1}, if s ≥3,

we nd that

⎧⎨π1 (G x 0 )



192 Chap.4 C

Theorem 4.3.4 Let M (n1, n2, · · ·, nm ) be a d-connected nitely coifold for an integer d, 1 ≤ d ≤ n1. If ∀(M 1, M 2) ∈E (GL [M

M 1 ∩M 2 is simply connected, then

(1) for ∀x0∈Gd, M ∈V (GL [M (n1, n2, · · ·, n m )]) and x0M

πd(M (n1, n 2, · · ·, nm ), x0)∼= (M ∈V (G d )

πd(M, x M 0)) π

where Gd = Gd[M (n1, n 2, · · ·, nm )] in which each edge (M 1, M 2)given point xM 1 M 2 ∈M 1 ∩M 2, πd(M, x M 0), π (Gd, x0) denote thgroups of a manifold M and the graph Gd respectively and



groups of a manifold M and the graph G , respectively and

(2) for ∀x, y∈M (n1, n 2, · · ·, nm ),

πd(M (n1, n2, · · ·, nm ), x)∼= πd(M (n1, n2, · · ·, nm ),

Proof Applying Corollary 3 .1.3, we rstly prove that the fundof two arcwise connected spaces S 1 and S 2 are equal if there exist subspaces U, V ⊂S 1, U, V ⊂S 2 such that U ∩V is simply co

U ∩V = {z0}in S 2, such as those shown in Fig.4 .3.1.

U ∩V U V U z0


for x0∈U ∩V and

π1(S 2, z0) = π1(U, z0)π1(V, z0)

by Corollary 3.1.3. Whence, π1(S 1, x0) = π1(S 2, z0). Therdetermine equivalently the fundamental d-group of a new coM ∗(n1, n2, · · ·, nm ), which is obtained by replacing each pM (n1, n 2,

· · ·, n m ) by M 1

∩M 2 =

{xM 1 M 2

}, such as those sh

X X



¹ Y

Z

Z M (n1, n2, · · ·, nm ) M ∗(n1, n2, ·

Fig. 4.3.2

For proving the conclusion (1), we only need to prove t

M (n1, n 2, · · ·, n m ), there are elements C M 1 , C M 2 , · · ·, C M l(M )∈π

∈π(Gd) and integers aM i , b j for∀M ∈V (Gd) and 1 ≤ i ≤

β (Gd) such that

C

≡M ∈V (Gd )

l(M )

i=1

aM i C M

i +c(G d )

j =1

b j α j (mod

194 Chap.4 C

{aM i |1 ≤ i ≤ l(M )}= {kM

i |kM i = 0 and 1 ≤ i ≤b(M

{b j |1 ≤ j ≤c(Gd)}= {l j |l j = 0 , 1 ≤ j ≤β (Gd)}Then we get that

C

≡M ∈V (Gd )

l(M )

i=1

aM

iC M

i+

c(G d )

j =1

b j

α j(mod2) . (3.4

The formula (3 .4.1) provides with us



[C ]∈(M ∈V (G d )

πd(M, x M 0)) π(Gd, x0).

If there is another decomposition

C ≡M ∈V (Gd )

l (M )

i=1

a M i C M

i +c (G d )

j =1

b j α j (mod2) ,

not loss of generality, assume l (M ) ≤ l(M ) and c (M ) ≤c(M ), t

M ∈V (Gd )

l(M )

i=1

(aM i −a M

i )C M i +

c(G d )

j =1

(b j −b j )α j = 0

where a M i = 0 if i > l (M ), b j = 0 if j > c (M ). Since C M

i , 1α j , 1 ≤ j ≤β (Gd) are bases of the fundamental group π(M ) and π

we must have



196 Chap.4 C

need to prove that f π is an isomorphism.

By denition, there is also a homotopic equivalence g : M

M (n1, n2, · · ·, nm ) such that gf identity : M (n1, n2, · · ·, n m ) →Thereby, gπ f π = ( gf )π = μ(identity )π :

πd(M (n1, n 2, · · ·, nm ), x) →π s (M (n1, n 2, · · ·, nm ),

where μ is an isomorphism induced by a certain d-path from x to

· · ·, n m ). Therefore, gπ f π is an isomorphism. Whence, f π is a m

gπ is an epimorphism.

Similarly, apply the same argument to the homotopy



fg identity : M (k1, k2, · · ·, kl) →M (k1, k2, · · ·, k

we get that f π gπ = ( fg )π = ν (identity ) pi :

πd(M (k1, k2, · · ·, kl), x) →π s (M (k1, k2, · · ·, kl), x

where ν is an isomorphism induced by a d-path from fg (x) to x inSo gπ is a monomorphism and f π is an epimorphism. Combining tus to conclude that f π : πd(M (n1, n 2, · · ·, n m ), x) →πd(M (k1, k2,isomorphism.

Corollary 4.3.7 If f : M (n1, n 2, · · ·, nm ) →M (k1, k2, · · ·, kl) is athen for any integer d, 1 ≤d ≤n1 and x∈M (n1, n 2, · · ·, nm ),

πd(M (n1, n2, · · ·, nm ), x)∼= πd(M (k1, k2, · · ·, kl), f (



198 Chap.4 C

=0≤ j<i ≤ p

(−1)i+ j σ ◦F i,p ◦F j,p − 1 +0≤ j<i ≤ p

(−1)i+ j − 1σ

= 0 .

Denote by Z p(S ) all p-cycles and B p(S ) all boundaries in C pis a subgroup of C p(S ) by denition. According to Theorem 4Im∂ p+1 ≤Ker∂ p. This enables us to get a chain complex ( C ; ∂ )

0 → · · · →C p+1 (S )∂ p +1

→C p(S )∂ p

→C p− 1(S ) → · ·Similarly, the pth singular homology group of S is dened to b



H p(S ) = Z p(S )/B p(S ) = Ker ∂ p/ Im∂ p+1 .

These singular homology groups of S are topological invarinext.

Theorem 4.4.2 If S is homomorphic to T , then H p(S ) is isomorany integer p ≥0.

Proof Let f : S →T be a continuous mapping. It induces f : C p(S ) →C p(T ) by setting f σ = f ◦σ for each singular pextend it linearly on C p(S ).

Notice that

f (∂σ) =

p

i=0 (−1)i

f ◦σ ◦F i,p .

Sec.4.4 Homology Groups of Combinatorial Manifolds

Furthermore, singular homology groups are homotopy infollowing result. For its proof, the reader is referred to [Mas2

Theorem 4.4.3 If f : S →T is a homopoty equivalence, this an isomorphism for each integer p ≥0.

Now we calculate homology groups for some simple spa

Theorem 4.4.4 Let S be a disjoint union of arcwise conne

and ι p : S λ →S an inclusion. Then for each p ≥ 0, the inH p(S λ ) →H P (S ) induce an isomorphism

H p(Sλ )(ιλ )∗

∼= H p(S)



λ∈Λ

H p(S λ ) ∼H p(S ).

Proof Notice that the image of a singular simplex mustconnected component of S . It is easily to know that each (introduced in the proof of Theorem 4 .4.2 induces isomorphi

λ∈ΛC p(S λ )

(ι λ )

∼= C p(S ),

λ∈Λ Z p(S λ )

(ι λ )

∼= Z p(S ),

λ∈ΛB p(S λ )

(ι λ )

∼= B p(S ).

Therefore, we know that

λ∈ΛH p(S λ )

(ιλ )∗

∼= H p(S ).

200 Chap.4 C

[π1(S, x 0), π1(S, x 0)] = a− 1b− 1ab|a, b∈π1(S, x 0)

Proof The ( i) is an immediately consequence of Theorem 4proof can be found in references, for examples, [Mas2], [You1], et

Theorem 4.4.6 Let O be a one point space. Then singular homare

H p(O) = Z, if p = 0 ,0, if p > 0.

Proof The case of p = 0 is a consequence of Theorem 4.4.4



there is exactly one singular simplex σ p : Δ p →O. Whence, each cis an innite cyclic group generated by σ p. By denition,

∂σ p = p

i=0

(−1)iσ p ◦F i,p = p

i=0

(−1)i σ p− 1 =0, if p

σ p− 1, if p

Therefore, ∂ : C p(O) →C p− 1(O) is an isomorphism if p is even an p is odd. We get that

· · ·∼=

→C 3(O) 0

→C 2(O)∼=

→C (O) 0

→C 0(O) →0.

By this chain complex, it follows that for each p > 0,

Z p(O) =C p(O), if p is odd,

0, if p is even;


It is easily to know also that the boundary operatorposses the property that ∂ p(C p(A))⊂C p(A). Whence, it ind

∂ p on quotient groups

∂ p : C p(S, A) →C p− 1(X, A).

Similarly, we dene the p-cycle group and p-boundary g

Z p(S, A) = Ker ∂ p =

{u

∈

C p(S, A)

|∂ p(u)

B p(S, A) = Im ∂ p+1 = ∂ p+1 (C p+1 (S, A)

for any integer p ≥0. Notice that ∂ p∂ p+1 = 0. It follows thd h h l i h l H (S A) i d d b



and the pth relative homology group H p(S, A) is dened to b

H p(S, A) = Z p(S, A)/B p(S, A).

Let (S, A) and (T, B) be pairs consisting of a topologicalA continuous mapping f : S →T is called a mapping (S, A) idenoted by f : (S, A) →(T, B) such a mapping.

The main property of relative homology groups is the e

in the following result. Its proof is refereed to the reference

Theorem 4.4.7 Let (S, A) be a pair and B a subset of A sucthe interior of A. Then the inclusion mapping i : (S −B, Aan isomorphism of relative homology groups

H p(S −B, A −B)i∗

∼= H p(S, A)

202 Chap.4 C

is called a short exact chain . Notice that the exactness of a short ethat ∂ 3 is surjective, Ker ∂ 3 = Im ∂ 4 and

C 2 ∼= C 3/ Ker∂ 3 = C 3/ Im∂ 4

by Theorem 2 .2.5.Now let i : A →S be an inclusion mapping for a pair ( S, A)

C p(S, A) the natural epimorphism of C p(S ) onto its quotient gr

an integer p ≥ 0. Then as shown in the proof of Theorem 4.4.homomorphisms i∗ : H p(A) →H p(S ), j∗ : H p(S ) →H p(S, A) for

We dene a boundary operator ∂ ∗ : H p(S, A) →H p− 1(A

∀u∈H p(S, A), choose a representative p-cycle u ∈C p(S, A) for



∀∈ p( ) p p y ∈ p( )is an epimorphism, there is a chain u”∈C p(S ) such that j (u”) =chain ∂ (u”). We nd that j ∂ (u”) = ∂j (u”) = ∂u = 0. Whencethe subgroup C p− 1(A) of C p− 1(S ). It is a cycle of C p(S, A). We dhomology class of the cycle ∂ (u”). It can be easily veried that ∂on the choice of u , u” and it is a homomorphism, i.e., ∂ ∗(u + v) =

∀u, v∈H p(S, A).Therefore, we get a chain complex, called the homology sequ

lowing.

· · ·j ∗

→H p+1 (S, A) ∂ ∗

→H p(A) i∗

→H p(S ) j ∗

→H p(S, A) ∂ ∗

→Theorem 4.4.8 The homology sequence of any pair (S, A) is exa

Proof It is easily to verify the following six inclusions:


Theorem 4.4.9(Mayer-Vietoris) Let S be a topological S 1∪S 2 = S . Then for each integer p ≥0, there is a homom

H p− 1(S 1 ∩S 2) such that the following chain

· · ·∂ ∗

→H p(S 1 ∩S 2)i∗⊕ j∗

→H p(S 1)⊕H p(S 2) k∗− l∗

→H p(S ) ∂ ∗

→H

is exact, where i∗⊕ j∗(u) = ( i∗(u), j∗(u)) , ∀u ∈H p(S 1 ∩S 2k∗(u)

−l∗(v) for

∀

u

∈

H p(S 1), v

∈

H p(S 2).

This theorem and the exact chain in it are usually calletheorem and Mayer-Vietoris chain , respectively. For its prooto [Mas2] or [Lee1].



[ ] [ ]

4.4.4 Homology Group of d-Dimensional Graph. W

fundamental group of d-dimensional graphs in Section 4 .3. Tin previous subsections also enables us to nd its singular ho

Theorem 4.4.10 For an integer n ≥1, the singular homoloare

H p(S n)∼=

Z, if p = 0 or n,0, otherwise.

Proof Let N and S denote the north and south poles of V = S n \ {S }. By the Mayer-Vietoris theorem, we know ththe Mayer-Vietoris chain

204 Chap.4 C

for p > 1 and n ≥1.Now if n = 1, H 0(S 1) ∼= H 1(S 1) ∼= Z by Theorem 4 .4.5

previous relation shows that H p(S 1

)∼= H p− 1(S 0

). Notice that Sisolated points. Applying Theorems 4 .4.5 and 4.4.6, we know thconsequently, H p(S 1) is a trivial group.

Suppose the result is true for S n − 1 for n > 1. The cases obtained by Theorem 4 .4.5. For cases of p > 1, applying the reagain, we nd that

H p(S n )∼= H p− 1(S n − 1)∼=⎧⎪⎪⎨⎪⎪⎩

0, if p < n,Z, if p = n,0 if p > n



⎪⎨⎪⎪⎩

0, if p > n.


Corollary 4.4.1 A sphere S n is not contractible to a point.

Corollary 4.4.2 The relative homology groups of the pair (Bn, S

H p(Bn, S n − 1)∼=

0, p = n,Z, p = n

for p, n ≥1.

Proof Applying Theorem 4 .4.8, we know an exact chain foll

· · ·→H p(B n ) j∗

→H p(B n , S n − 1) ∂ ∗

→H p− 1(S n − 1) i∗

→H p− 1(B

n


H p(ei , ei)

∼

=0, p = n,

Z, p = n,where ei∼= B n and ei = ei −ei∼= S n − 1 for integers 1 ≤ i ≤Theorem 4.4.11 Let M d(G) be a d-dimensional graph with EThen the inclusion (el , el) →(M d(G), V (M d(G))) induces a mH p(M d(G), V (M d(G))) for l = 1 , 2

· · ·, m and H p(M d(G),

sum of the image subgroups, which follows that

H p(M d(G), V (M d(G)))∼=⎧⎪⎨⎪⎩

Z⊕ · · ·Z

m

, if

0 if



⎧⎪⎨⎪⎩ 0, if

Proof For a ball Bd

and the sphere S d− 1

with center D d

12

= {x∈R d| x ≤12}. Let f l : B d →el be a continuou

1 ≤ l ≤m in the space of M d(G) and

D l = f l(D d12), al = f l(0), A = {a l|1 ≤ l ≤

X = M d

(G) \ A, D =

m

l=1 D l .

Notice that f l maps a pair ( D d, D d−{0}) homeomorphicand those subsets D l , 1 ≤ l ≤ m are pairwise disjoint. Wediagram

H p(D , D

−A) 1

→H p(M d(G), X ) 2

←H p(M d(G), M d(G

h h d t h hi i d d b th

206 Chap.4 C

Consequently, H p(M d(G), V (M d(G))) = 0 if p = d and H d(M d(Gfree Abelian group with basis in 1 −1 correspondent with the set M

Consider the following diagram:

H p(D , D −A) ¹ H p(M d(G), X ) H p(M d(G),

H p(Dd

, Dd

− {0})

¹

H p(Bd

, Bd

− {0})

H p(Bd

1 2

3 4

f l∗ f ”l∗

The vertical arrows denote homomorphisms induced by f l .maps (D d, D d − {0}) homeomorphically onto ( D l, D l − {a l}).

H (D d D d 0 ) i hi ll h di d H



maps H p(D d, D d−{0}) isomorphically onto the direct summand HH p(D , D −A). We have proved that arrows 1 and 2 are isomorphis

the same method we can also know that arrows 3 and 4 are isomoring all these facts suffices to know that f l∗ : H p(B

d, S d− 1) →H p(M

is a monomorphism. This completes the proof.Particularly, if d = 1, i.e., M d(G) is a graph G embedded in a

we know its homology groups in the following.

Corollary 4.4.3 Let G be a graph embedded in a topological spac

H p(G, V (G))∼=⎧⎪⎨

⎪⎩

Z⊕ · · ·Z

ε(G)

, if p = 1 ,

0, if p = 1 .

Corollary 4.4.4 Let X = M d(G), X v = V(M d(G)). Then th


4.4.5 Homology Group of Combinatorial Manifold.determining homology groups of combinatorial manifolds is

balls to a d-dimensional graph, i.e., there exists a d-dimenM (n1, n 2, · · ·, n m ) such that

M (n1, n2, · · ·, nm ) \ M d[G] =k

i=2

li

j =1

B

where B i j is the i-ball B i for integers 1

≤i

≤k, 1

≤ j

≤li .

result for homology groups of combinatorial manifolds.

Theorem 4.4.12 Let M be a combinatorial manifold, M d(Ggraph with E (M d(G)) = {e1, e2, · · ·, em}such that



{ }M

\M d[G] =

k

i=2

li

j =1

B i j .

Then the inclusion (el , el) →(M, M d(G)) induces a monoH p(M, M d(G)) for l = 1 , 2 · · ·, m and

H p(M, M d(G))∼=⎧

⎪⎨⎪⎩

Z⊕ · · ·Z

m

, if p =

0, if p =

Proof Similar to the proof of Theorem 4 .4.11, we can g

Corollary 4.4.5 Let M be a combinatorial manifold, M d(Ggraph with E (M d(G)) = {e1, e2, · · ·, em}such that

M \ M d[G] =k li

B i j .

208 Chap.4 C

Theorem 4.4.13 For any manifold in a combinatorial manifold chain

· · ·j ∗

→H p+1 (M, M ) ∂ ∗

→H p(M ) i∗

→H p(M ) j ∗

→H p(M, M

is exact.

Proof It is an immediately conclusion of Theorem 4 .4.8.

For a nitely combinatorial manifold, if each manifold in thmanifold is compact, we call it a compactly combinatorial manifohomology groups of compactly combinatorial manifolds following.

Theorem 4.4.14 A compact combinatorial manifold M is nitel



Proof It is easily to know that the homology groups H p(M ) o

natorial manifold M can be generated by [u]∈H p(M )|M ∈V (Ga famous result, i.e., any compact manifold is nitely generated (tails), we know that M is also nitely generated.

§4.5 REGULAR COVERING OF

COMBINATORIAL MANIFOLDS BY VOLTAGE ASSIGN

4.5.1 Action of Fundamental Group on Covering Space.a covering mapping of topological spaces. For ∀x0∈S , the set p−

bre over the vertex x0, denoted by b x ) . Notice that we have ii 1( ) 1( ) i h f f h 3 1 12

Sec.4.5 Regular Covering of Combinatorial Manifolds

lifted arc Ll over L starting at x.

Notice that L : bx →by is a bijection by the proof

∀C ∈π1(M ), let L∗= L− 1CL. Then

(L, L∗) : (bx , π1(S, p(x))) →(bx , π1(S, p

is an isomorphism of actions.

4.5.2 Regular Covering of Labeled Graph. Generalizion graphs in topological graph theory ([GrT1]) to vertex-edge us to nd a combinatorial technique for getting regular covmanifold M , which is the essence in the construction of pri



combinatorial manifolds in follow-up chapters.Let GL be a connected vertex-edge labeled graph with

of a label set and Γ a nite group. A voltage labeled graph ograph GL is a 2-tuple (GL ; α) with a voltage assignments α

α(u, v) = α− 1(v, u ), ∀(u, v)∈E (GL

Similar to voltage graphs such as those shown in Examplof voltage labeled graphs lies in their labeled lifting GL α de

V (GL α ) = V (GL ) ×Γ, ( u, g)∈V (GL ) ×Γ abbre

E (GLα ) = {(ug, vg◦ h ) | for ∀(u, v)∈E (GL ) with

with labels Θ L : GL α

→L following:

210 Chap.4 C

p− 1(u, v) = {(ug, vg◦ h ) | ∀g∈Γ }for an edge (u, v)∈E (GL ) with α(u, v) = h. Such sets p− 1(u), p

bres over the vertex u∈V (GL ) or edge (u, v)∈E (GL ), denotedrespectively.

A voltage labeled graph with its labeled lifting are shown in FiGL = C L3 and Γ = Z 2.

3

4

12

2 5

3

3

54

4

2

2

11

2

2



4 2(GL , α )

5 4

GL α

2

Fig. 4.5.1

A mapping g : GL →GL is acting on a labeled graph GθL : GL →L if gθL (x) = θL g(x) for∀x∈V (GL )∪E (GL ), and aon a labeled graph GL if each g

∈

Γ is acting on GL . Clearly, i

labeled graph GL , then Γ ≤ Aut G. In this case, we can dene graph GL / Γ by

V (GL / Γ) = {uΓ | ∀u∈V (GL ) },

E (GL / Γ) = {(u, v)Γ | ∀(u, v)∈E (GL )}and a labeling θΓL : GL / Γ →L with


A group Γ is freely acting on a labeled graph GL if for∀element in V (GL )∪E (GL ) implies that g is the unit eleme

every element in GL

.For voltage labeled graphs, a natural question is whiclifting of a voltage labeled graph (GL , α ) with α : E (GL )question, we introduce an action Φ g of Γ on GL α for∀g∈Γ

For ∀g∈Γ, the action Φg of g on GLα is dened by Φg

ΘL Φg, where ΘL : GL α

→L is the labeling on GL α

induced bThen we know the following criterion.

Theorem 4.5.2 Let Γ be a group acting freely on a labelequotient graph GL / Γ Then there is an assignment α : E (G



quotient graph G / Γ. Then there is an assignment α : E (Gof vertices in GL by elements of V (GL )

×Γ such that GL =

the given action of Γ on GL is the natural left action of Γ o

Proof By denition, we only need to assign voltages onthe existence of a assignment such that GL = GL α , withouton these element in GL and GL already existence.

For this object, we choose positive directions on edges ofquotient mapping qΓ : GL →GL is direction-preserving and GL preserves directions rst. Then, for for each vertex v inof the orbit q− 1

Γ (v) in GL by v1Γ and for every group elementhe vertex φg(v1Γ ) as vg. Now if the edge e of GL runs fromlabel eg to the edge of orbit q− 1

Γ (e) that originates at the ve

freely on GL , there are just |Γ| edges in the orbit q− 1Γ (e), on

212 Chap.4 C

Under this relabeling process, the isomorphism ϑ : GL →GL α

GL with bers of GL α . Moreover, it is dened precisely so that tGL is consistent with the natural left action of Γ on the lifting gra

The construction of lifting from a voltage labeled graph impresult, which means that GL α is a |Γ|-fold covering over (GL , α ) wi

Theorem 4.5.3 Let GL α be the lifting of the voltage labeled α : E (GL ) →Γ. Then

|bu | = |b(u,v ) | = |Γ| for ∀u∈V (GL ) and (u, v)∈E

and furthermore, denote by C vGL (l) and C eGL (l) the sets of vertilabel l∈L in a labeled graph GL Then



label l∈L in a labeled graph G . Then

|C vGL α (l)| = |Γ||C vGL (l)| and |C eGL α (l)| = |Γ||C eGL (l

Proof By denition, Γ is freely acting on GL α . Whence, we

|b(u,v )| = |Γ| for∀u∈V (GL ) and (u, v)∈E (GL ). Then it follow

|Γ||C vGL (l)| and |C eGL α (l)| = |Γ||C eGL (l)|.4.5.3 Lifting Automorphism of Voltage Labeled Graph.action of the fundamental group of GL , we can nd criterions Lft( f ) of a automorphism f ∈Aut GL . First, we have two generaon the lifting automorphism of a labeled graph.

Theorem 4.5.4 Let p : GL

→GL be a covering projection and f

of GL Then f lifts to a f l∈Aut GL if and only if for an arbitr


where L : p(u) →u is an arc.

Proof First, let f l be a lifting of f and L : p(u) →u

f l(u) →f l(u ·L) projects to f (L), which implies that f lParticularly, this equality holds for ∀u∈bu and L∈π1(Gthe required isomorphism of action is obtained.

Conversely, let ( ϕ, f ) be such an isomorphism. We denan arbitrary vertex v in GL and v = p(v). Let L : v →u be

f l(v) = ϕ(v ·f (L− 1)) .

Then this mapping is well dened, i.e., it does not depend fact let L1 L2 : v u Then v L1 = ( v L2) L− 1L1



fact, let L1, L2 : v →u. Then v ·L1 = ( v ·L2) ·L2 L1.ϕ((v

·L2))

·f (L− 1

2 L1) = ϕ((v

·L2))

·f (L− 1

2 )

·f (L1). There

L1) ·f (L− 11 ) = ϕ(v ·L2) ·f (L− 1

2 ).

From the denition of f l it is easily seen that pf l(v) is a bijection. First, we show it is onto. Now let w be an aand choose L : p(w) →f (u) arbitrarily. Then it is easily toϕ

− 1(w ·L) · f − 1(L− 1) mapped to w. For its one-to-one, le

f l(v1) = f l(v2) = ϕ(v2 ·L2) ·f (L− 12 ). Whence, f (L1) and f (Lvertex. Consequently, so do L1 and L2. Therefore, v1 andFurthermore, we know that ϕ(v1 ·L1) ·f (L− 1

1 L2) = ϕ(v2 ·ϕ(v1 ·L1 ·L− 1

1 L2) = ϕ(v2 ·L2). That is, ϕ(v1 ·L2) = ϕ(v2 ·L2

and so v1 = v2.

Now we conclude that f l

is really a lifting of f . Thi




α(W ) = identity ⇒ α(f (W )) = identi

and locally f -invariant for an automorphism f ∈Aut GL

with respect to the group f in Aut GL

. Notice that for eacsatisfying the required inference. Whence, the local A-invariarequirement that for ∀f ∈A, there exists an induced isomorpof local voltage groups such that the following diagram

π1(GL , u) π1(GL , f ¹

f

f # u

α



π1(GL , u) π1(GL , f ¹

f

Fig. 4.5.2

is commutative, i.e., f # u (α(W )) = α(f (W )) for ∀W ∈π1(Gcriterion for lifting automorphisms of voltage labeled graphs.

Theorem 4.5.6 Let (GL , α ) be a voltage labeled graph wif ∈Aut GL . Then f lifts to an automorphism of GL α if and f -invariant.

Proof By denition, the mapping ( lu , α ) : (bu , π1(Glu : bu →Γ is a bijection. Whence, if W ∈π1(GL , uW

∈

(π1(GL , u))u if and only if α(W )

∈

Γug , i.e., gα(W ) =

α(W) = identity

216 Chap.4 C

The voltage assignment technique on the labeled graph GL [M a combinatorial manifold M ∗ by Theorem 4 .2.4. Assume (GLα [Mof GL [M ] with α : E (GL [M ])

→Γ. For

∀

M

∈

V (GL [M ]), let hself-homeomorphism of M , ς M : x →M for∀x∈M , and deneThen we know that p∗ : M ∗→M is a covering projection.

Theorem 4.5.7 (M ∗, p∗) is a |Γ|-sheeted covering, called natural

Proof For M ∈V (GL [M ]), let x∈M . By denition, for ∀and ∀h

− 1s (x)∗∈M g, we know that

p∗((h− 1s (x))∗) = hs ◦ς − 1

M g pς M g ((h− 1s (x))∗) = hs (h− 1

s (x)) =

By denitions of the voltage labeled graph and the mapping



By denitions of the voltage labeled graph and the mappingily that each arcwise component of ( p∗)− 1(U x ) is mapped topolneighborhood U x for∀x∈M . Whence, p∗ : M ∗→M is a coverin

Notice that there are |Γ| copies M g , g ∈Γ for ∀M ∈V (G(M ∗, p∗) is a |Γ|-sheeted covering of M .

Let p1 : S 1 →S and p2 : S 2 →S be two covering projectispaces. They are said to be equivalent if there exists a one-to-one S 2 such that the following

S 1 S 2¹

τ

p1 p2


Theorem 4.5.8 The number nc(M ) of non-equivalent naturacombinatorial manifold M is

nc(M ) = 1

|Aut |GL[M]g∈Aut GL [M ]

|Φ(g)

where Φ(g) = {α : E (GL ) →Γ|αg = gα}.

Proof B denition, two voltage labeled graphs ( GL [M

equivalent if there is an one-to-one mapping f : V (GL

[M ])fα = αf and fθ L = θL f . Whence, there must be that f ∈ACorollary 2.5.4, we get the conclusion.

Particularly, if Aut GL [M ] is trivial or transitive, we gf h i l l i f i bi



for the non-equivalent natural covering of a nite combinator

Corollary 4.5.1 Let M be a nitely combinatorial manifold

(i) if AutGL [M ] is trivial, then

nc(M ) = ε|Γ |(GL [M ]).

(ii ) if AutGL [M ] is transitive, then

nc(M ) = |Γ|+ ε(GL [M ]) −1ε(GL [M ])

.

Proof If AutGL [M ] is trivial, then α : E (GL [M ])

in GL [M ] and such mappings induce non-equivalent natural

218 Chap.4 C

As a part of enumerating non-equivalent natural coverings, mcians turn their attentions to non-equivalent surface coverings of a with a trivial voltage group Γ. Such as those of results in [Mao1], [[Mul1] and [MRW1]. For example, if GL [M ] is the labeled comphave the following result in [Mao1] for surface coverings.

Theorem 4.5.9 The number nc(M ) with GL [M ]∼= K Ln , n ≥5 on

nc(M ) = 12

(k |n

+k |n,k ≡ 0(mod 2)

) 2α (n,k )(n −2)!nk

knk ( n

k )!+

k |(n − 1),k =1

φ(k)2

where,

n (n − 3) if k 1(mod2);



α(n, k ) = 2k , if k ≡1(mod2);n (n − 2)

2k , if k ≡0(mod2),and

β (n, k ) =(n − 1)( n − 2)

2k , if k ≡1(mod2);(n − 1)( n − 3)

2k , if k ≡0(mod2).

and nc(M ) = 11 if G

L[M ]∼= K

L4 .

For meeting the needs of combinatorial differential geometry iters, we introduce the conception of combinatorial ber bundles f

Denition 4.5.2 A combinatorial ber bundle is a 4-tuple (M ∗, Mof a covering combinatorial manifold M ∗, a group G, a combinatand a projection mapping p : M∗→M with properties following:

Sec.4.6 Remarks

Theorem 4.5.10 Let M be a nite combinatorial manifold anage labeled graph with α : E (GL ([M ]) →Γ. Then (M ∗, M, p ber bundle, where M ∗ is the combinatorial manifold correGL α ([M ], p∗ : M ∗→M a natural projection determined by hs : M →M a self-homeomorphism of M and ς M : x →Mς M (x) = M for ∀x∈M .

§4.6 REMARKS

4.6.1 How to visualize a Euclidean space of dimension≥one hard to understand. Certainly, we can describe a point of



clidean space R n by an n-tuple ( x1, x2, · · ·, xn ). But how to

since one can just see objects in R 3. The combinatorial Eucliapproach decomposing a higher dimensional space to a lowera combinatorial structure. The discussion in Section 4 .1 mpacking problem, i.e., in what conditions do R n 1 , R n 2 , · · ·, Rnatorial Euclidean space E G (n1, n2, · · ·, nm )? Particularly, the

problem.

Problem 4.6.1 Let R n 1 , R n 2 , · · ·, R n m be Euclidean spacessional number dimE G (n1, n2, · · ·, nm ), particularly, the dimensr ≥2 for a given graph G.

Theorems 4 .1.1

−4.1.3 partially solved this problem, and

got the number dim E K (r ) But for any connected graph

220 Chap.4 C

heartening thing in Section 4 .2 is the correspondence of a combwith a vertex-edge labeled graph, which enables one to get its reSection 4.5 and combinatorial elds in Chapter 8.

4.6.3 The well-known Seifer and Von Kampen theorem on fundavery useful in calculation of fundamental groups of topological sits application to a wide range, the following problem is interestin

Problem 4.6.2 Generalize the Seifer and Von-Kampen theorem tU ∩V maybe not arcwise connected.

Corollary 4.3.4 is an interesting result in combinatorics, whicfundamental group of a surface can be completely determined by aon this surface. Applying this result to enumerate rooted or unroot



maps on surfaces (see [Mao1], [Liu2] and [Liu3] for details) ithrough inquiry.

4.6.4 Each singular homology group is an Abelian group by deniwe always nd singular groups of a space with the form of Z ×4.4.11−4.4.12 determined the singular homology groups of combin

constraint on conditions. The reader is encourage to solve the gesingular homology groups of combinatorial manifolds following.

Problem 4.6.3 Determine the singular homology groups of combin

Furthermore, the inverse problem following.

Problem 4.6.4 For an integer n ≥ 1, determine what kind of



Sec.5.1 Differentiable Combinatorial Manifolds

§5.1 DIFFERENTIABLE COMBINATORIAL MANIFOLD

5.1.1 Smoothly Combinatorial Manifold. We introducon nitely combinatorial manifolds and characterize them in

Denition 5.1.1 For a given integer sequence 1 ≤ n1 < nbinatorial C h -differential manifold (M (n1, n2, · · ·, nm ); A)rial manifold M (n1, n 2, · · ·, nm ), M (n1, n2, · · ·, nm ) =

i∈I U

A= {(U α ;ϕα )|α∈I }on M (n1, n2, · · ·, nm ) for an integer h following hold.

(1) {U α ; α∈I }is an open covering of M (n1, n 2, · · ·, n

(2) For ∀α, β ∈I , local charts (U α ;ϕα ) and (U β ;ϕU α U β = ∅or U α U β = ∅but the overlap maps



∅ ∅

ϕαϕ− 1β : ϕβ (U α U β ) →ϕβ (U β ) and ϕβ ϕ

− 1α : ϕα (U α

are C h -mappings, such as those shown in Fig. 5.1.1 followin

¹

U α

U β

U α ∩U β

ϕα

ϕ

ϕ

ϕβ ϕ− 1α




A1 = {(V x ;ϕx )|∀x∈M 1} A2 = {(W y; ψy)|∀ysuch that ϕx |V x W y = ψy|V x W y for ∀x ∈M 1, y∈M 2, thestructures

A= {(U p; [ p])|∀ p∈M (n1, n2, · · ·, nm

such that (M (n1, n 2,

· · ·, n m );

A) is a combinatorial C h -diffe

Proof By denition, We only need to show that we can borhood U p and a homoeomorphism [ p] for each p∈M (nthese conditions (1) −(3) in denition 3 .1.

By assumption, each manifold ∀M ∈V (Gd[M (n1, n2, · ·



accordingly there is an index set I M such that

{U α ; α

∈

I of M , local charts (U α ;ϕα ) and (U β ;ϕβ ) of M are equivalis maximal. Since for ∀ p∈M (n1, n 2, · · ·, nm ), there is a l

p such that [ p] : U p →s ( p)

i=1B n i ( p) , i.e., p is an intersect

M n i ( p) , 1 ≤ i ≤ s( p). By assumption each manifold M n i ( p)

exists a local chart ( V i

p ;ϕi p) while the point p∈M

n i ( p)

such we dene

U p =s ( p)

i=1

V i p .

Then applying the Gluing Lemma again, we know that there [ ] U h h

226 Chap.5 Combinatorial D

5.1.2 Tangent Vector Space. For a point in a smoothly combiwe introduce the tangent vector at this point following.

Denition 5.1.3 Let (M (n1, n2, · · ·, nm ), A) be a smoothly comband p∈M (n1, n2, · · ·, nm ). A tangent vector v at p is a mappingconditions following hold.

(1) ∀g, h∈X p,∀λ∈R , v(h + λh ) = v(g) + λv(h);

(2) ∀g, h∈X p, v(gh) = v(g)h( p) + g( p)v(h).

Let γ : (− , ) →M be a smooth curve on M and p = γ (0). Twe usually dene a mapping v : X p →R by

v(f ) =df (γ (t))

dt |t=0 .



We can easily verify such mappings v are tangent vectors at p.Denoted all tangent vectors at p∈M (n1, n 2, · · ·, nm ) by T p

and dene addition¡ + ¢ and scalar multiplication ¡ ·¢ for∀u, v∈T pMλ∈R and f ∈

X p by

(u + v)(f ) = u(f ) + v(f ), (λu )(f ) = λ ·u(f ).

Then it can be shown immediately that T pM (n1, n 2, · · ·, nm ) is a vthese two operations ¡ + ¢ and¡ ·¢ . Let

X (M (n1, n 2, · · ·, nm )) = p∈M

T pM (n1, n2, · · ·, nm

A vector eld on M (n1, n 2, · · ·, nm ) is a mapping X : M →X (M

i e chosen a vector at each point p∈M (n1 n2 nm )


for X,Y,Z ∈X (M (n1, n2, · · ·, nm )) ,

(i) [X, Y ] = −[Y, X ];

(ii ) the Jacobi identity [X, [Y, Z ]] + [Y, [Z, X ]] + [Z, [X, Y ]] =

holds. Such systems are called Lie algebras.

For ∀ p ∈M (n1, n 2, · · ·, nm ), We determine the dimen

tangent space T pM (n1, n 2, · · ·, nm ) in the next result.Theorem 5.1.3 For any point p∈M (n1, n 2, · · ·, nm ) withthe dimension of T pM (n1, n 2, · · ·, nm ) is

dimT pM (n1, n2, · · ·, nm ) = s( p) +s ( p)

i=1(n i −



i 1

with a basis matrix

[∂

∂x]s ( p)× n s ( p ) =

⎡⎢⎢⎢⎢⎢⎣

1s( p)

∂ ∂x 11 · · · 1

s( p)∂

∂x 1 s ( p )∂

∂x 1( s ( p )+1) · · · ∂ ∂x 1n 1

1s( p)

∂ ∂x 21 · · · 1

s( p)∂

∂x 2 s ( p )∂

∂x 2( s ( p )+1) · · · ∂ ∂x 2n 2

· · · · · · · · · · · · · · · ·1

s( p)∂

∂x s ( p )1 · · · 1s( p)

∂ ∂x s ( p ) s ( p )

∂ ∂x s ( p )( s ( p )+1) · · · · · · ∂x s

where xil

= x jl

for 1 ≤ i j ≤ s(p) 1 ≤ l ≤ s(p) name


{

∂

∂xhj

| p

|1

≤ j

≤ s( p)

}(

s ( p)

i=1

n i

j = s( p)+1 {

∂

∂xij

| p

|1

≤ j

≤s

})

for a given integer h, 1 ≤h ≤s( p), namely (5 −1) is a basis of T pMIn fact, for ∀x∈[ϕ p](U p), since f is smooth, we know that

f (x) −f (x0) =

1

0

ddt f (x0 + t(x −x0))dt

=s( p)

i=1

n i

j =1

η j

s ( p) (xij −xij

0 )1

0

∂ f ∂x ij (x0 + t(x

in a spherical neighborhood of the point p in [ϕ ](U ) R s(p)− s(



in a spherical neighborhood of the point p in [ϕ p](U p)

⊂

R

s( p) s(

with [ϕ p]( p) = x0, where

η j

s ( p) =1

s ( p) , if 1≤ j ≤ s( p),1, otherwise .

Dene

gij (x) =1

0

∂ f ∂x ij (x0 + t(x −x0))dt

and gij = gij ·[ϕ p]. Then we nd that

( ) ( ) ∂f ( )


v(f ( p)) = 0 , and v(η j

s( p)x

ij0 ) = 0 .

Accordingly, we obtain that

v(f ) = v(f ( p) +s( p)

i=1

n i

j =1

η j

s( p) (xij −xij

0 )gij ( p))

= v(f ( p)) +

s ( p)

i=1

n i

j =1 v(η j

s( p)(xij

−xij0 )gij ( p)))

=s ( p)

i=1

n i

j =1

(η j

s( p)gij ( p)v(xij −xij0 ) + ( xij ( p) −xij

0 )v

=s ( p) n i

η js (p)

∂f ∂ ij ( p)v(xij )



i=1 j =1

η

s ( p) ∂x ij (p) ( )

=s ( p)

i=1

n i

j =1

v(xij )η j

s( p)∂

∂x ij | p(f ) = [vij ]s( p)× n s ( p ) , [


v = [vij ]s ( p)× n s ( p ) , [ ∂ ∂x ]s ( p)× n s ( p ) . (5 −The formula (5 −2) shows that any tangent vector v

can be spanned by elements in (5 .1).Notice that all elements in (5 −1) are also linearly ind

there are numbers a ij , 1

≤i

≤s( p), 1

≤ j

≤n i such that


By Theorem 5.1.3, if s( p) = 1 for any point p∈(M (n1, n 2,dimT pM (n1, n 2, · · ·, n m ) = n1. This can only happens while Mcombined by one manifold. As a consequence, we get a well-knowndifferential geometry again.

Corollary 5.1.1 Let (M n ;A) be a smooth manifold and p∈M n

dimT pM n = n

with a basis

{∂

∂x i | p | 1 ≤ i ≤n}.

5.1.3 Cotangent Vector Space. For a point on a smoothmanifold, the cotangent vector space is dened in the next deniti



, g p

Denition 5.1.5 For ∀ p∈(M (n1, n 2, · · ·, nm ); A), the dual spaceis called a co-tangent vector space at p.

Denition 5.1.6 For f ∈X p, d∈T ∗ p M (n1, n2, · · ·, nm ) and v∈T

the action of d on f , called a differential operator d : X p →R , is

df = v(f ).

Then we immediately obtain the result following.

Theorem 5.1.4 For ∀ p∈(M (n1, n 2, · · ·, n m ); A) with a local cdimension of T ∗ p M (n1, n 2, , n m ) is

Sec.5.2 Tensor Fields on Combinatorial Manifolds

where xil = x jl for 1 ≤ i, j ≤s( p), 1 ≤ l ≤ s( p), namely for aat a point p of M (n1, n 2, · · ·, nm ), there is a smoothly functiosuch that,

d = [uij ]s ( p)× n s ( p ) , [dx]s( p)× n s ( p ) .

§5.2 TENSOR FIELDS ON COMBINATORIAL MANIFO

5.2.1 Tensor on Combinatorial Manifold. For any intof type (r, s ) at a point in a smoothly combinatorial manifoldened following.

Denition 5.2.1 Let M (n1, n 2, · · ·, nm ) be a smoothly comp M (n1 n 2 nm ) A tensor of type (r s ) at the point



p

∈

M (n1, n 2,

· · ·, nm ). A tensor of type (r, s ) at the point

is an (r + s)-multilinear function τ ,

τ : T ∗ p M ×· · ·×T ∗ p M

r×T pM ×· · ·×T pM

swhere T pM = T pM (n1, n 2,

· · ·, nm ) and T ∗ p M = T ∗ p M (n1, n 2

Denoted by T rs ( p, M ) all tensors of type ( r, s ) at a pointThen we know its structure by Theorems 5 .1.3 and 5.1.4.

Theorem 5.2.1 Let M (n1, n 2, · · ·, nm ) be a smoothly com p∈M (n1, n2, · · ·, nm ). Then


t = t i1 ··· i r j1 ··· j s

∂ ∂x i1 j 1 | p⊗ · · ·⊗

∂ ∂x i r j r | p⊗dxk1 l1

⊗ · ·for components t i1 ··· i r

j 1 ··· j s ∈R by Theorems 5 .1.3 and 5.1.4, where 1 ≤1 ≤ jh ≤ ih , 1 ≤ lh ≤kh for 1 ≤h ≤ r . As a consequence, we obt

T rs ( p,M ) = T pM ⊗ · · ·⊗T pM

r

⊗T ∗ p M ⊗ · · ·⊗T ∗ p M

s

Since dimT pM = dim T ∗ p M = s( p) +s ( p)

i=1(n i − s( p)) by Theorem

we also know that

dimT rs ( p, M ) = ( s( p) +s ( p)

i=1(n i −s( p))) r + s .



5.2.2 Tensor Field on Combinatorial Manifold. Similarcan also introduce tensor eld and k-forms at a point in a combifollowing.

Denition 5.2.2 Let T rs (M ) =

p∈M

T rs ( p,M ) for a smoothly comb

M = M (n1, n 2, · · ·, nm ). A tensor led of type (r, s ) on M (nmapping τ : M (n1, n 2, · · ·, n m ) →T rs (M ) such that τ ( p) ∈TM (n1, n2, · · ·, nm ).

A k-form on M (n1, n2, · · ·, nm ) is a tensor eld ω∈T 0k (M ). Dof M (n

1, n

2, , n

m) by Λk(M ) and


Aω(u1, · · ·, uk) =1k!

σ∈S k

signσω(uσ(1) , · · ·,

for ∀u1 ∈M , and for integers k, l ≥ 0 and ω∈Λk(M ),ω∧ ∈Λk+ l(M ) is dened by

ω∧ =(k + l)!

k!l!A (ω⊗ ).

For example, if M = R 3, a is a 1-form and b a 1-form,

a∧b (e1, e2) = a(e1)b (e2) −a(e2)b (e

and if a is a 2-form and b a 1-form, then



a∧b (e1, e2, e3) = a(e1, e2)b (e3) −a(e1, e3)b (e2) +

Example 5.2.1 The wedge product is operated in Λ( M ) inalgebraic case. For example, let a = dx1

−x1dx2

∈

Λ1(M ) dx2∧dx1∈Λ2(M ), then

a∧b = ( dx1 −x1dx2)∧(x2dx1∧dx3 −dx2

= 0 x1x2dx2

∧

dx1

∧

dx3 dx1

∧

dx2

= ( x x 1)∧dx dx ∧dx ∧dx


v1∧v2∧ · · ·∧vn

v1∧ · · ·∧vn − 1

∧(a1v1 + a2v2 · · ·+ an − 1vn − 1

= 0 .

Now if v1, v2, · · ·, vn are linear independent, we can extend

{v1, v2, · · ·, vn , · · ·, vdim V }of V . Because of

v1∧v2∧ · · ·∧vdim V = 0 ,

we nally get that

v1∧v2∧ · · ·∧vn = 0 .



Theorem 5.2.3 Let v1, v2, · · ·, vn and w1, w2, · · ·, wn be two vvector space V such that

n

k=1

vk∧wk = 0 .

If v1, v2,

· · ·, vn are linear independent, then for any integer k, 1

≤wk =

n

l=1

akl vl

with akl = a lk .

Proof Because v1, v2, , vn are linear independent, we can basis {v v v vd }of V Therefore there are scala


=n

1≤ k<l ≤ n

(akl −a lk )vk∧vl +n

k=1

dim V

t= k+

by assumption. Since {vk∧vl , 1 ≤k < l ≤dim

V

}is a basiakl −a lk = 0 and akt = 0. Thereafter, we get that

wk =n

l=1

akl vl

with akl = a lk .

5.2.3 Exterior Differentiation. It is the same as in thgeometry, the next result determines a unique exterior differeΛ(M ) for smoothly combinatorial manifolds.

Theorem 5.2.4 Let M be a smoothly combinatorial man



unique exterior differentiation d : Λ(M ) →Λ(M ) such thatd(Λk)⊂Λk+1 (M ) with conditions following hold.(1) d is linear, i.e., for ∀ϕ, ψ∈Λ(M ), λ∈R ,

d(ϕ + λψ ) = dϕ∧ψ + λdψ

and for ϕ

∈

Λk(M ), ψ

∈

Λ(M ),

d(ϕ∧ψ) = dϕ + ( −1)kϕ∧dψ.

(2) For f ∈Λ0(M ), df is the differentiation of f .

(3) d2 = d ·d = 0 .

(4) d is a local operator i e if U⊂V⊂M are open se


dα =∂α (μ1 ν 1 )··· (μk ψk )

∂x μν dxμν ∧dxμ1 ν 1

∧ · · ·∧dxμk ν k . (5

and d is uniquely determined on U by properties (1) −(3) and bsubset of M .

For existence, dene on every local chart ( U ; [ϕ]) the operator(2) is trivially veried as is R -linearity. If β = β (σ1 ς 1 )··· (σl ς l )dxσ1 ς 1

∧then

d(α∧β ) = d(α (μ1 ν 1 )··· (μk ψk )β (σ1 ς 1 )··· (σ l ς l )dxμ1 ν 1∧ · · ·∧dxμ k ν k

∧dx

= (∂α (μ1 ν 1 )··· (μk ψk )

∂x μν β (σ1 ς 1 )··· (σl ς l ) + α (μ1 ν 1 )··· (μk ψk )

×∂β (σ1 ς 1 )··· (σl ς l )

∂x μν )dxμ1 ν 1∧ · · ·∧dx μ k ν k

∧dxσ1 ς 1∧ · · ·d



= ∂α (μ1 ν 1 )··· (μk ψk )∂x μν dxμ1 ν 1

∧ · · ·∧dxμ k ν k

∧β (σ1 ς 1 )··· (σ l ς l )dx

+ ( −1)kα (μ1 ν 1 )··· (μk ψk )dxμ1 ν 1 · · ·∧dxμ k ν k

∧∂β (σ1 ς 1 )··· (σl ς l

∂x μν

= dα∧β + ( −1)kα∧dβ

and (1) is veried. For (3), symmetry of the second partial derivat

d(dα) =∂ 2α (μ1 ν 1 )··· (μk ψk )

∂x μν ∂x σς dxμ1 ν 1∧ · · ·∧dx μ k ν k

∧dxσ1 ς 1∧ · ·

Thus, in every local chart ( U ; [ϕ]), (5 −3) denes an operator dIt remains to be shown that d really denes an operator d on anyholds To do so it suffices to show that this denition is chart ind


dM (ϕ + λψ ) = dM ϕ + λdM ψ.

(2) For ϕ

∈

Λr (M ), ψ

∈

Λ(M ),

dM (ϕ∧ψ) = dM ϕ + ( −1)rϕ∧dM ψ.

(3) For f ∈Λ0(M ), dM f is the differentiation of f .

(4) d2M = dM ·dM = 0 .

Then d|M = dM .

Proof By Theorem 2.4.5 in [AbM1], dM exists uniquelyifold M . Now since d is a local operator on M , i.e., for anyd(α|U μ ) = ( dα)|U μ and there is an index set J such that M



| |that

d|M = dM

by the uniqueness of d and dM .

Theorem 5.2.5 Let ω

∈

Λ1(M ). Then for

∀

X, Y

∈

X (M )

dω(X, Y ) = X (ω(Y )) −Y (ω(X )) −ω([X,

Proof Denote by α(X, Y ) the right hand side of the fα : M ×M →C ∞ (M ). It can be checked immediately tha

∀X, Y ∈X (M ), f ∈C ∞ (M ),


α|U = dω|U .

In fact, assume ω|U = ωμν dxμν . Then

(dω)|U = d(ω|U ) =∂ωμν

∂x σς dxσς ∧dxμν

=12

(∂ωμν

∂x σς −∂ωςτ

∂x μν )dxσς ∧dxμ

On the other hand, α

|U = 1

2 α( ∂ ∂x μν , ∂

∂x σς )dxσς

∧

dxμν , where

α(∂

∂x μν ,∂

∂x σς ) =∂

∂x σς (ω(∂

∂x μν )) −∂

∂x μν (ω(∂

∂x σ

−ω([∂

∂x μν −∂

∂x σς ])

=∂ωμν ∂ωσς .



∂x σς −∂x μν

Therefore, dω|U = α|U .

§5.3 CONNECTIONS ON TENSORS

5.3.1 Connection on Tensor. We introduce connections on tecombinatorial manifolds by the next denition.

Denition 5.3.1 Let M be a smoothly combinatorial manifold. tensors of M is a mapping D : X (M ) T rs M T rs M with DX

that for ∀X Y ∈X M τ π∈T r (M ) λ∈R and f ∈C∞ (M )




As a consequence, we get that ( DX 1 τ 1)V = ( DX 1 τ 2)V , particul(DX 1 τ 2) p. For the arbitrary choice of p, we get that ( DX 1 τ 1)U = (

The local property of D enables us to nd an induced connectT rs (U ) →T rs (U ) such that D U

X |U (τ |U ) = ( DX τ )|U for∀X ∈

X (MNow for∀X 1, X 2∈

X (M ), ∀τ 1, τ 2∈T rs (M ) with X 1|V p = X 2|V p

dene a mapping D U : X (U ) ×T rs (U ) →T rs (U ) by

(DX 1 τ 1)|V p = ( DX 1 τ 2)|V p

for any point p∈U . Then since D is a connection on M , it can be cD U satises all conditions in Denition 5 .3.1. Whence, D U is inon U .

Now we calculate the local form on a chart ( U p, [ϕ p]) of p. S

D ∂ Γ κλ(σς )( μν )∂



D ∂∂x μν = Γ κλ(σς )( μν ) ∂x σς ,

it can nd immediately that

D ∂∂x μν

dxκλ = −Γκλ(σς )( μν ) dxσς

by Denition 5.3.1. Therefore, we nally nd that

(DX τ )|U = X σς τ (μ1 ν 1 )( μ2 ν 2 )··· (μ r ν r )(κ1 λ 1 )( κ2 λ 2 )··· (κ s λ s ),(μν )

∂ ∂x μ1 ν 1 ⊗ · · ·⊗

∂ ∂x μ r ν r ⊗dxκ

with

∂τ (μ1 ν1 )( μ2 ν2 )··· (μ r ν r )( λ )( λ ) ( λ )

Sec.5.3 Connections on Tensors

T (X, Y ) = DX Y −DY X −[X, Y ]

is a tensor of type (1, 2) on M .

Proof By denition, it is clear that T : X (M ) ×X

symmetrical and bilinear. We only need to check it is also linC ∞ (M ) for variables X or Y . In fact, for ∀f ∈C ∞ (M ),

T (fX,Y ) = D fX Y −DY (fX ) −[fX,Y= f DX Y −(Y (f )X + f DY X

− (f [X, Y ]−Y (f )X ) = f T (X

and



T (X,fY ) = −T (fY,X ) = −f T (Y, X ) = f T

5.3.2 Torsion-free Tensor. Notice that

T (∂

∂x μν ,∂

∂x σς ) = D ∂∂x μν

∂ ∂x σς −D ∂

∂x σς ∂

= (Γ κλ(μν )( σς ) −Γκλ

(σς )( μν ) )

under a local chart ( U p; [ϕ p]) of a point p ∈M . If T ( ∂ ∂x μν

torsion-free . This enables us getting the next useful result by

Theorem 5 3 3 A connection D on tensors of a smoothly


Z (g(X, Y )) = g(DZ , Y ) + g(X, DZ Y ) (5 −4)

then M is called a combinatorial Riemannian geometry, denoted b

We get a result for connections on smoothly combinatorial mathat of the Riemannian geometry.

Theorem 5.3.4 Let (M, g) be a combinatorial Riemannian maniexists a unique connection D on (M, g) such that (M,g, D) is a c

mannian geometry.Proof By denition, we know that

DZ g(X, Y ) = Z (g(X, Y )) −g(DZ X, Y ) −g(X, DZ

for a connection D on tensors of M and

∀

Z

∈

X (M ). Thereby, this equivalent to that of D g = 0 for ∀Z∈

X (M ) namely D is to



∀∈is equivalent to that of DZ g = 0 for ∀Z ∈X (M ), namely D is to

Not loss of generality, assume g = g(μν )( σς ) dxμν dxσς in a locala point p, where g(μν )( σς ) = g( ∂

∂x μν , ∂ ∂x σς ). Then we nd that

Dg = (∂g(μν )( σς )

∂xκλ

−g(ζη )( σς ) Γζη

(μν )( σς )

−g(μν )( ζη )Γζη

(σς )( κλ ) )dxμν

⊗Therefore, we get that

∂g(μν )( σς )

∂x κλ = g(ζη )( σς ) Γζη(μν )( σς ) + g(μν )( ζη )Γζη

(σς )( κλ ) (5 −if DZ g = 0 for ∀Z ∈

X (M ). The formula (5 −5) enables us to g

Sec.5.4 Curvatures on Connections Spaces

Accordingly, D = D∗. Whence, there are at most one torsiona combinatorial Riemannian manifold ( M, g).

For the existence of torsion-free connection D on (M, g)

and dene a connection D on (M, g) such that

D ∂∂x μν

= Γ κλ(σς )( μν )

∂ ∂x κλ ,

then D is torsion-free by Theorem 5 .3.3. This completes the

Corollary 5.3.1 For a Riemannian manifold (M, g), there free connection D , i.e.,

DZ g(X, Y ) = Z (g(X, Y )) −g(DZ X, Y ) −g(X,

for ∀X,Y,Z ∈X (M ).



§5.4 CURVATURES ON CONNECTION SPACES

5.4.1 Combinatorial Curvature Operator. A combina

is a 2-tuple (M, D) consisting of a smoothly combinatorial mnection D on its tensors. We dene combinatorial curvature ocombinatorial manifolds in the next.

Denition 5.4.1 Let (M, D) be a combinatorial connectioX (M ), a combinatorial curvature operator R(X, Y ) : X (M

b


(1) R(X, Y ) = −R(Y, X );

(2) R(fX,Y ) = R(X,fY ) = f R(X, Y );

(3) R(X, Y )(fZ ) = f R(X, Y )Z .Proof For ∀X,Y,Z ∈

X (M ), we know that R(X, Y )Z denition. Whence, R(X, Y ) = −R(Y, X ).

Now since

R(fX,Y )Z = D fX DY Z −DY D fX Z −D [fX,Y ]Z

= f DX DY Z −DY (f DX Z ) −D f [X,Y ]− Y

= f DX DY Z −Y (f )DX Z −f DY DX Z

− f D [X,Y ]Z + Y (f )DX Z

= f R(X Y )Z



f R(X, Y )Z,

we get that R(fX,Y ) = f R(X, Y ). Applying the quality (1), we

R(X,fY ) = −R(fY,X ) = −f R(Y, X ) = f R(X, Y

This establishes (2). Now calculation shows that

R(X, Y )(fZ ) = DX DY (fZ ) −DY DX (fZ ) −D [X,Y ](fZ

= DX (Y (f )Z + f DY Z ) −DY (X (f )Z + f

− ([X, Y ](f ))Z −f D [X,Y ]Z

X (Y(f ))Z + Y(f )D Z + X (f )D Z




5.4.2 Curvature Tensor on Combinatorial Manifold.5.4.1, the curvature operator R(X, Y ) : X (M ) →X (M ) isBy applying this operator, we can dene a curvature tensor i

Denition 5.4.2 Let (M, D) be a combinatorial connectionX (M ), a linear multi-mapping R: X (M )×X (M )×X (Mby

R(Z,X,Y ) =

R(X, Y )Z

is said a curvature tensor of type (1, 3) on (M, D).

Let (M, D) be a combinatorial connection space and

{eij

|1

≤i

≤s( p), 1

≤ j

≤n i and ei1 j = ei2 j for 1

≤i1, i2

≤



{ | ≤ ≤ ≤ ≤ ≤ ≤a local frame with a dual

{ωij |1 ≤ i ≤s( p), 1 ≤ j ≤n i and ωi1 j = ωi2 j for 1 ≤ i1, i2 ≤abbreviated to {eij }and {ωij }at a point p∈M , where MThen there exist smooth functions Γ σς

(μν )( κλ )∈C ∞ (M ) such

D eκλ eμν = Γ σς (μν )( κλ ) eσς

called connection coefficients in the local frame

{eij

}.


with Rκλ(μν )( σς )( σς ) eκλ = R(eσς , eηθ )eμν .

Proof By denition, for any given eσς , eηθ we know that

(dωμν −ωκλ∧ωμν

κλ )(eσς , eηθ) = eσς (ωμν (eηθ )) −eηθ (ωμν (eσς ))

− ωκλ (eσς )ωμν κλ (eηθ) + ωκλ (eηθ )ωκ

= −ωμν σς (eηθ) + ωμν

ηθ (eσς ) −ωμν ([

=

−Γμν

(σς )( ηθ) + Γ μν (ηθ)( σς )

−ωμν ([eσ

= ωμν (Deσς eηθ −Deηθ eσς −[eσς ,

= ωμν (T (eσς , eηθ)) = T μν (σς )( ηθ) .

by Theorem 5 .2.3. Whence,

dωμν

−ωκλ

∧ωμν κλ =

12T

μν (κλ )( σς )ω

κλ

∧ωσς

.



∧ ∧Now since

(dωκλμν −ωϑι

μν ∧ωκλϑι )(eσς , eηθ)

= eσς (ωκλμν (eηθ )) −eηθ (ω

κλμν (eσς )) −ω

κλμν ([eσς , eηθ ])

−ωϑιμν (eσς )ωκλ

ϑι (eηθ) + ωϑιμν (eηθ )ωκλ

ϑι (eσς )

= eσς (Γκλ(μν )( ηθ) ) −eηθ (Γκλ

(μν )( σς ) ) −ωϑι ([eσς , eηθ])Γκ(

−Γϑι(μν )( σς ) Γ

κλ(ϑι )( ηθ) + Γ ϑι

(μν )( ηθ) Γκλ(ϑι )( σς )

and


(dωκλμν −ωϑι

μν ∧ωκλϑι )(eσς , eηθ) = Rκλ

(μν )( σς )(

that is,

dωκλμν −ω

σς μν ∧ω

κλσς =

12 R

κλ(μν )( σς )( ηθ) ω

σς

∧ω

5.4.3 Structural Equation. First, we introduce torsion fand structural equations in a local frame {eij }of (M, D) in

Denition 5.4.3 Let (M, D) be a combinatorial connecti2-forms Ωμν = dωμν −ωμν

∧ωμν κλ , Ωκλ

μν = dωκλμν −ωσς

μν ∧ωκλσς a

dωμν = ωκλ∧ωμν

κλ + Ω μν , dωκλμν = ωσς

μν ∧ωκλσς

are called torsion forms, curvature forms and structural equa

{eij }of (M, D), respectively.



{ }By Theorem 5.4.3 and Denition 5.4.3, we get local f

and curvature tensor in a local frame following.

Corollary 5.4.1 Let (M, D) be a combinatorial connection frame with a dual

{ωij

}at a point p

∈

M . Then

T = Ωμν ⊗eμν and R = ωμν

⊗eκλ ⊗Ω

i.e., for ∀X, Y ∈X (M ),

T(X Y ) Ωμν

(X Y ) d R(X Y ) Ωκλ

(X


dΩμν = −dωμν ∧ωμν

κλ + ωμν ∧dωμν

κλ

= −(Ωκλ

+ ωσς

∧ωκλσς )∧ω

μν κλ + ω

κλ

∧(Ωμν κλ + ω

σκ

= ωκλ∧Ωμν

κλ −Ωκλ∧ωμν

κλ .

Similarly, differentiating the equality Ω κλμν = dωκλ

μν −ωσς μν ∧ωκλ

σς on also nd that

dΩκλμν = ωσς μν ∧Ωκλσς −Ωσς μν

∧ωκλσς .

Corollary 5.4.2 Let (M, D ) be an affine connection space and with a dual {ωi}at a point p∈M . Then

dΩi = ω j∧Ωi j −Ω j

∧ωi j and dΩ ji = ωki ∧Ω jk −Ωki



∧ ∧ ∧

5.4.4 Local Form of Curvature Tensor. According to 5.4.4 there is a type (1, 3) tensor R p : T pM ×T pM ×T pM →T p

R(w,u,v ) = R(u, v)w for∀u,v,w∈T pM at each point p∈M . P

its a concrete local form in the standard basis {∂

∂x μν }.Theorem 5.4.5 Let (M, D) be a combinatorial connection space. with a local chart (U p; [ϕ p]),

R= Rηθ(σς )( μν )( κλ ) dxσς

⊗∂

∂x ηθ ⊗dxμν ⊗dxκλ

i h


R(∂

∂x μν ,∂

∂x κλ )∂

∂x σς = Rηθ(σς )( μν )( κλ )

∂ ∂x η

at the local chart ( U p

; [ϕ p

]). In fact, by denition we get tha

R(∂

∂x μν ,∂

∂x κλ )∂

∂x σς

= D ∂∂x μν

D ∂∂x κλ

∂ ∂x σς −D ∂

∂x κλD ∂

∂x μν

∂ ∂x σς −D [ ∂

∂x μν , ∂∂x κλ ] ∂

= D ∂∂x μν (Γηθ(σς )( κλ )

∂ ∂x ηθ ) −D ∂

∂x κλ (Γηθ(σς )( μν )∂

∂x ηθ )

=∂ Γηθ

(σς )( κλ )

∂x μν ∂

∂x ηθ + Γ ηθ(σς )( κλ ) D ∂

∂x μν

∂ ∂x ηθ −

∂ Γηθ(σς )( μν )

∂x κλ∂

∂x η

= (∂ Γηθ

(σς )( κλ )

∂x μν −∂ Γηθ

(σς )( μν )

∂x κλ )∂

∂x ηθ + Γ ηθ(σς )( κλ )Γ

ϑι(ηθ)( μν )

∂ ∂xϑι

= (∂ Γηθ

(σς )( κλ )∂x μν −

∂ Γηθ

(σς )( μν )∂x κλ + Γ ϑι(σς )( κλ ) Γηθ(ϑι )( μν ) −Γϑι(σς )( μν ) Γ



∂ ∂

= Rηθ(σς )( μν )( κλ )

∂ ∂x ηθ .


For the curvature tensor

Rηθ(σς )( μν )( κλ ) , we can also get

in the next result.

Theorem 5.4.6 Let (M, D) be a combinatorial connection swith a local chart (U p, [ϕ p]), if T ≡0, then

Rμν

( λ )( )( θ) Rμν

( )( θ)( λ ) Rμν

( θ)( λ )( )


Rμν (κλ )( σς )( ηθ) + Rμν

(σς )( ηθ)( κλ ) + Rμν (ηθ)( κλ )( σς )

= R(∂

∂x σς ,∂

∂x ηθ )∂

∂x κλ + R(∂

∂x ηθ ,∂

∂x κλ )∂

∂x σς + R(∂

∂x κλ , ∂

with

X =∂

∂x σς , Y =∂

∂x ηθ and Z =∂

∂x κλ

in the rst Bianchi equality and

Dϑι Rκλ(μν )( σς )( ηθ) + Dσς Rκλ

(μν )( ηθ)(ϑι ) + DηθRκλ(μν )(ϑι )( σς )

= Dϑι R(∂

∂x σς ,∂

∂x ηθ )∂

∂x κλ + Dσς R(∂

∂x ηθ ,∂

∂xϑι )∂

∂x κλ + DηθR

= 0 .

ith



with

X =∂

∂xϑι , Y =∂

∂x σς , Z =∂

∂x ηθ , W =∂

∂x κλ

in the second Bianchi equality of Theorem 5 .4.2.

§5.5 CURVATURES ON RIEMANNIAN MANIFOLDS

5.5.1 Combinatorial Riemannian Curvature Tensor. In th

b l f ld d h

Sec.5.5 Curvatures on Riemannian Manifolds

for ∀X,Y,Z,W ∈X (M ).

Then we nd symmetrical relations of R(X,Y,Z,W ) fo

Theorem 5.5.1 Let R : X (M ) ×X (M ) ×X (M ) ×Xcombinatorial Riemannian curvature tensor. Then for ∀X,Y

(1) R(X,Y,Z,W ) + R(Z,Y,W,X ) + R(W,Y,X,Z ) = 0

(2) R(X,Y,Z,W ) = −R(Y,X,Z,W ) and R(X,Y,Z,W

(3) R(X,Y,Z,W ) = R(Z,W,X,Y ).Proof For the equality (1), calculation shows that

R(X,Y,Z,W ) + R(Z,Y,W,X ) + R(W,Y,X,

= g(R(Z, W )X, Y ) + g(R(W,X )Z, Y ) + g(R

= g(R(Z, W )X + R(W,X )Z + R(X, Z )W,Y



g( ( , ) ( , ) ( , ) ,

by denition and Theorem 5 .4.1(4).For (2), by denition and Theorem 5 .4.1(1), we know t

R(X,Y,Z,W ) = g(R(Z, W )X, Y ) = g(−R(W

= −g(R(W,Z )X, Y ) = −R(X

Now since D is a combinatorial Riemannian connection,


g(DW DZ X, Y ) = W (Z (g(X, Y ))) −W (g(X, DZ

− Z (g(X, DW Y )) + g(X, DZ DW

Notice that

g(D [Z,W ], Y ) = [Z, W ]g(X, Y ) −g(X, D [Z,W ]Y ).

By denition, we get that

R(X,Y,Z,W ) = g(DZ DW X −DW DZ X −D [Z,W ]X, Y )

= g(DZ DW X, Y ) −g(DW DZ X, Y ) −g(D

= Z (W (g(X, Y )))

−Z (g(X, DW Y ))

−W (

+ g(X, DW DZ Y ) −W (Z (g(X, Y ))) + W (



+ Z (g(X, DW Y )) −g(X, DZ DW Y ) −[Z, W

− g(X, D [Z,W ]Y )

= Z (W (g(X, Y ))) −W (Z (g(X, Y ))) + g(X

− g(X, DZ DW Y ) −[Z, W ]g(X, Y ) −g(X,= g(X, DW DZ Y −DZ DW Y + D [Z,W ]Y )

= −g(X, R(Z, W )Y ) = −R(Y,X,Z,W ).

Applying the equality (1), we know that

Sec.5.5 Curvatures on Riemannian Manifolds

by applying (2). We also know that

R(W,Y,X,Z )

−R(X,Z,Y,W ) =

−(R(Z,Y,W,X )

= R(X,Y,Z,W ) −This enables us getting the equality (3)

R(X,Y,Z,W ) = R(Z,W,X,Y ).

5.5.2 Structural Equation in Riemannian Manifold.5.4.2 −5.4.3 and 5.5.1, we also get the next result.

Theorem 5.5.2 Let (M,g, D) be a combinatorial RiemannianΩσς

μν g(σς )( κλ ) . Then

(1) Ω(μν )( κλ ) = 12 R (μν )( κλ )( σς )( ηθ) ωσς ∧ωηθ;

(2) Ω(μν )( κλ ) + Ω (κλ )( μν ) = 0;



(2) Ω(μν )( κλ ) + Ω (κλ )( μν ) 0;(3) ωμν

∧Ω(μν )( κλ ) = 0;(4) dΩ(μν )( κλ ) = ωσς

μν ∧Ω(σς )( κλ ) −ωσς κλ ∧Ω(σς )( μν ) .

Proof Notice that T ≡0 in a combinatorial Riemanni

We nd that

Ωκλμν =

12

Rκλ(μν )( σς )( ηθ) ωσς

∧ωηθ

by Theorem 5 .4.2. By denition, we know that


5.5.3 Local form of Riemannian Curvature Tensor. Forwith a local chart ( U p, [ϕ p]), we can also nd a local form of R in

Theorem 5.5.3 Let R :X

(M ) ×X

(M ) ×X

(M ) ×X

(M )combinatorial Riemannian curvature tensor. Then for ∀ p∈M (U p; [ϕ p]),

R = R(σς )( ηθ)( μν )( κλ ) dxσς ⊗dxηθ

⊗dxμν ⊗dxκλ

with

R (σς )( ηθ)( μν )( κλ ) =12

(∂ 2g(μν )( σς )

∂x κλ ∂x ηθ +∂ 2g(κλ )( ηθ)

∂x μνν ∂x σς −∂ 2g(μν )( ηθ)

∂x κλ ∂x σς

+ Γ ϑι(μν )( σς ) Γ

ξo(κλ )( ηθ)g(ξo)(ϑι ) −Γξo

(μν )( ηθ) Γ(κλ )( σς

where g(μν )( κλ ) = g(∂

∂x μν ,∂

∂x κλ ).Proof Notice that



Proof Notice that

R(σς )( ηθ)( μν )( κλ ) = R(∂

∂x σς ,∂

∂x ηθ ,∂

∂x μν ,∂

∂x κλ ) = R(∂

∂x μν ,∂

∂x

= g(R(∂

∂x σς ,∂

∂x ηθ )∂

∂x μν ,∂

∂x κλ ) = Rϑι(μν )( σς )( ηθ

By denition and Theorem 5 .5.1(3). Now we have know that (eq

∂g(μν )( κλ )

∂x σς = Γ ηθ(μν )( σς ) g(ηθ)( κλ ) + Γ ηθ

(κλ )( σς )g(μν )( ηθ) .

Sec.5.6 Integration on Combinatorial Manifolds

+Γ ϑι(σς )( μν ) (Γ

ξo(ϑι )( κλ )g(ξo)( ηθ) + Γ ξo

(ηθ)( κλ ) g(ϑι )( ξo)) + Γ ξo(σς )

−Γϑι(σς )( κλ ) (Γ

ξo(ϑι )( μν ) g(ξo)( ηθ) + Γ ξo

(ηθ)( μν ) g(ϑι )( ξo)) −Γξo(σς )

=1

2

∂

∂x μν (∂g(σς )( ηθ)

∂x κλ+

∂g(κλ )( ηθ)

∂x σς −∂g(σς )( κλ )

∂x ηθ) + Γ ξo

(σς

−12

∂ ∂x κλ (

∂g(σς )( ηθ)

∂x μν +∂g(μν )( ηθ)

∂x σς −∂g(σς )( μν )

∂x ηθ ) −Γξo(σς )

=12

(∂ 2g(μν )( σς )

∂x κλ ∂x ηθ +∂ 2g(κλ )( ηθ)

∂x μν ∂x σς −∂ 2g(μν )( ηθ)

∂x κλ ∂x σς −∂ 2g(κλ )

∂x μν ∂x+Γ ξo

(σς )( κλ ) Γϑι(ξo)( μν ) g(ϑι )( κλ ) −Γξo

(σς )( μν ) Γϑι(ξo)( κλ ))g(ϑι )( ηθ)

This completes the proof.Combining Theorems 5 .4.6, 5.5.1 and 5.5.3, we have the

Corollary 5.5.1 Let R (μν )( κλ )( σς )( ηθ) be a component of a comcurvature tensor R in a local chart (U, [ϕ]) of a combinatoria

(1) R (μν )( κλ )( σς )( ηθ) = −R (κλ )( μν )( σς )( ηθ) = −R (μν )( κλ )( ηθ)( σ(2) R (μν )( κλ )( σς )( ηθ) = R (σς )( ηθ)( μν )( κλ ) ;



(3) R (μν )( κλ )( σς )( ηθ) + R (ηθ)( κλ )( μν )( σς ) + R (σς )( κλ )( ηθ)( μν ) =(4) Dϑι R (μν )( κλ )( σς )( ηθ) + Dσς R (μν )( κλ )( ηθ)(ϑι ) + DηθR (μν )(

§5.6 INTEGRATION ON COMBINATORIAL MANIFOLD

5.6.1 Determining H M (n , m ). Let M (n1, · · ·, nm ) be a s

manifold. Then there exists an atlas C = {(U α , [ϕα ])|α ∈consisting of positively oriented charts such that for ∀α∈I ,

H


Theorem 5.6.1 Let M be a smoothly combinatorial manifold witvertex-edge labeled graph G([0, nm ], [0, nm ]). Then

H M (n, m )⊆ {n1, n 2, · · ·, nm}

d( p)≥ 3,p∈M { d( p) +

d( p)

i=1

(n i −d

{τ 1(u) + τ 1(v) −τ 2(u, v)|∀(u, v)∈E (G([0,

Particularly, if G([0, nm ], [0, nm ]) is K 3-free, then

H M (n, m ) = {τ 1(u)|u∈V (G([0, nm ], [0, nm ]))}

{τ 1(u) + τ 1(v) −τ 2(u, v)|∀(u, v)∈E (G([0,

Proof Notice that the dimension of a point p

∈

M is

n p = d( p) +d( p)

(n i −d( p))



p (p)i=1

( (p))

by denition. If d( p) = 1, then n p = n j , 1 ≤ j ≤ m. If d p∈M n i ∩M n j for 1 ≤ i, j ≤m, we know that its dimension is

n i + n j

−d( p) = τ 1(M n i ) + τ 1(M n j )

−d( p).

Whence, we get that

H M (n, m )⊆ {n1, n 2, · · ·, nm}

d( p)≥ 3,p∈M {d( p) +

d( p)

i=1

(n i −d


For some special graphs, we get the following interestingset H

M (n, m ).

Corollary 5.6.1 Let M be a smoothly combinatorial manifovertex-edge labeled graph G([0, nm ], [0, nm ]). If G([0, nm ], [0,

H M (n, m ) = {τ 1(ui), 1 ≤ i ≤ p} {τ 1(u i) + τ 1(u i+1 ) −τ 2(u

and if G([0, n m ], [0, n m ])∼= C p with p ≥4, then

H M

(n, m ) =

{τ 1(ui), 1

≤i

≤ p

} {τ 1(u i) + τ 1(u i+1 )

−τ 2(ui , u i+

5.6.2 Partition of Unity. A partition of unity on a comis dened following.

Denition 5.6.1 Let M be a smoothly combinatorial manisupport set Suppω of ω is dened by

Suppω = {p∈M ; ω(p) = 0}



Suppω = { p∈M ; ω( p) = 0}and say ω has compact support if Suppω is compact in M .

{C i|i∈I }of M is called locally nite if for each p∈M , theof p such that U p

∩C i =

∅

except for nitely many indices i

Denition 5.6.2 A partition of unity on a combinatorial man

{(U i , gi)|i∈I }, where

(1) {U i|i∈I }is a locally nite open covering of M ;(2) gi∈

X (M ), gi( p) ≥0 for∀ p∈M and supp gi∈U i


on M with conditions following hold.

(1) {U αM |α∈I M }is a locally nite open covering of M ;

(2) gαM ( p)

≥0 for

∀

p

∈

M and supp gαM

∈

U αM for α

∈

I M ;

(3) For p∈M ,i

giM ( p) = 1.

By denition, for ∀ p∈M , there is a local chart ( U p, [ϕ p]) B n i 1 B n i 2 · · · B n i s ( p ) with B n i 1 B n i 2 · · · B n i s ( p ) = ∅. N

· · ·, U αM i s ( p )be s( p) open sets on manifolds M, M ∈V (GL [M ]) su

p∈U α p =s( p)

h=1

U αM i h. (5 −8)

We dene

S ( p) =

{U α

p |all integers α enabling (5

−8) hold

Then



A= p∈M

S ( p) = {U α p |α∈I ( p)}

is locally nite covering of the combinatorial manifold M by prFor ∀U α p ∈S ( p), dene

σU αp =s≥ 1 {i1 ,i 2 ,··· ,i s }⊂{1,2,··· ,s ( p)}

(s

h=1

gM ςi h)

and


subordinate to A, i.e., for ∀i∈J , there exists α(i) such th

∀ p∈M ,

t( p) =i

gi tα (i)

is a C k tensor eld of type (r, s ) on M

Proof Since {U i|i∈J }is locally nite, the sum at eachand t( p) is a type (r, s ) for every p∈M . Notice that t is C k

t in a local chart ( V α (i) , [ϕα (i) ]) is

j

gi tα ( j ) ,

where the summation taken over all indices j such that Vnumber j is nite by the local niteness.

5.6.3 Integration on Combinatorial Manifold. First, w



on combinatorial Euclidean spaces. Let R (n1, · · ·, nm ) be a cspace and

τ : R (n1,

· · ·, nm )

→R (n1,

· · ·, nm )

a C 1 differential mapping with

[y] = [yκλ ]m × n m = [τ κλ ([xμν ])]m × n m .

The Jacobi matrix of f is dened by


Denoted by n = m +m

i=1(n i −m). If 0 ≤ l ≤ n, recall([4]

Λl(R (n1, · · ·, nm )) is

{e i1

∧ei2

∧ · · ·∧e i l |1 ≤ i1 < i 2 · · ·< i l ≤n}for a basis e1, e2, · · ·, en of R (n1, · · ·, nm ) and its dual basis e1, ethe dimension of Λl(R (n1, · · ·, nm )) is

nl =

(

m +

m

i=1(n i −m))!

l!( m +m

i=1(n i −m) −l)! .

Whence Λn (R (n1, · · ·, nm )) is one-dimensional. Now if ω0 is a bthen know that its each element ω can be represented by ω =c

∈

R . Let τ : R (n1,

· · ·, n m )

→R (n1,

· · ·, n m ) be a linear mappi

τ ∗ : Λn (R (n1, · · ·, nm )) →Λn (R (n1, · · ·, n m ))



( ( )) → ( ( ))

is also a linear mapping with τ ∗ω = cτ ∗ω0 = bω for a unique ccalled the determinant of τ . It has been known that ([AbM1])

det τ = det( ∂ [y]∂ [x])

for a given basis e1, e2, · · ·, en of R (n1, · · ·, nm ) and its dual basis

Denition 5.6.3 Let R (n1, n 2, · · ·, nm ) be a combinatorial Euc

m +m

i=1(n i −m), U ⊂R (n1, n 2, · · ·, nm ) and ω∈Λn (U ) have com


For example, consider the combinatorial Euclidean spaceR . Then the integration of an ω∈Λ7(U ) for an open subse

U ω = U ∩(R 3∪R 5 ) ω(x)dx1dx

12dx

13dx

22dx

23d

Theorem 5.6.3 Let U and V be open subsets of R (n1, · ·an orientation-preserving diffeomorphism. If ω∈Λn (V ) ha

n = m +m

i=1(n i −m), then τ ∗ω∈Λn (U ) has compact suppor

τ ∗ω = ω.

Proof Let ω(x) = ω(μ i 1 ν i 1 )··· (μ i n ν i n )dxμ i 1 ν i 1∧ · · ·∧dxμ i n

a diffeomorphism, the support of τ ∗ω is τ − 1(supp ω), whichsupp ω compact.

By the usual change of variables formula, since τ ∗ω = (nition, where ω0 = dx1

∧ · · ·∧dxm∧dx1(m +1)

∧dx1(m +2)∧



∧ ∧ ∧ ∧ ∧we then get that

τ ∗ω =

(ω

◦τ )(det τ )

m

ν =1

dxν

μ≥ m +1 ,1≤ ν

= ω.

Denition 5.6.4 Let M be a smoothly combinatorial man


Now for any integer n∈H

M (n, m ), we can dene an integrsmoothly combinatorial manifold M (n1, · · ·, nm ).

Denition 5.6.5 Let M be a smoothly combinatorial manifold wand (U ; [ϕ]) a positively oriented chart with a constant nU ∈

H

ω∈Λn U (M ), U ⊂M has compact support C ⊂U . Then dene

C ω = ϕ∗(ω|U ). (5 −10)

Now if C M is an atlas of positively oriented charts with an integ

let P = {(U α ,ϕα , gα )|α ∈I }be a partition of unity subordinate tΛn (M ), n∈

H M (n, m ), an integral of ω on P is dened by

P ω =

α∈I gα ω. (5 −11)

The following result shows that the integral of n-forms for∀well-dened.



Theorem 5.6.4 Let M (n1, · · ·, nm ) be a smoothly combinatorin ∈

H M (n, m ), the integral of n-forms on M (n1, · · ·, nm ) is we

the sum on the right hand side of (4.4) contains only a nite n

terms, not dependent on the choice of C M and if P and Q are

unity subordinate to C M , then

P ω = Q

ω.


Now let P = {(U α ,ϕα , gα )|α∈I }and Q = {(V β ,ϕβ , hof unity subordinate to atlas C

M and C ∗M

with respective inteH ∗

M (n, m ). Then these functions {gα hβ }satisfy gα hβ ( p) = 0

number of index pairs ( α, β ) and

α β

gα hβ ( p) = 1 , for∀ p∈M (n1, · · ·,

Sinceβ

= 1, we then get that

P =

α gα ω =β α hβ gα ω =

α β

By the relation of smoothly combinatorial manifolds w

labeled graphs established in Theorem 4 .2.4, we can also gvertex-edge labeled graph G([0, nm ], [0, nm ]) by viewing it th



smoothly combinatorial manifold M with Λl(G) = Λ l(M ), H

namely dene the integral of an n-form ω on G([0, nm ], [0,by

G([0,n m ],[0,n m ])ω = M

ω.

Then each integration result on a combinatorial manifold canbinatorial words, such as Theorem 5 .7.1 and its corollaries i

Now let n1, n2, , nm be a positive integer sequence. F


Similar to Theorem 5 .6.3 for the change of variables formucombinatorial Euclidean space, we get that of formula in smoothmanifolds.

Theorem 5.6.5 Let M (n1, n 2, · · ·, nm ) and N (k1, k2, · · ·, kl) betorial manifolds and τ : M →N an orientation-preserving diffω∈Λk(N ), k∈

H N (k, l) has compact support, then τ ∗ω has com

ω =

τ ∗ω.

Proof Notice that supp τ ∗ω = τ − 1(supp ω). Thereby τ ∗ω hassince ω has so. Now let {(U i ,ϕi)|i∈I }be an atlas of positively oriand P = {gi|i∈I }a subordinate partition of unity with an integeThen {(τ (U i),ϕi ◦ τ − 1)|i ∈I } is an atlas of positively orientedQ =

{gi

◦τ − 1

}is a partition of unity subordinate to the covering

an integer set H τ (M )(k, l). Whence, we get that



τ ∗ω =i giτ ∗ω =

i ϕi∗(giτ ∗ω)

=i

ϕi∗(τ − 1)∗(gi

◦τ − 1)ω

=i (ϕi ◦τ − 1)∗(gi ◦τ − 1)ω

= ω.

Sec.5.7 Combinatorial Stokes’ and Gauss’ Theorems

Class 1(interior point Int D ) For ∀ p∈Int D, there is aenable V p⊂D.

Case 2(boundary ∂ D ) For ∀ p ∈∂ D , there is integers

(U p; [ϕ p]) of p such that xμν ( p) = 0 but

U p ∩D = {q|q∈U p, xκλ ≥0 for ∀{κ, λ}=

Then we generalize the famous Stokes’ theorem on manibinatorial manifolds in the next.

Theorem 5.7.1 Let M be a smoothly combinatorial manifH

M (n, m ) and D a boundary subset of M . For ∀n∈H

M (na compact support, then

D dω = ∂ D ω

with the convention ∂D ω = 0 , while ∂ D = ∅.



∂ D ∅Proof By Denition 5.6.5, the integration on a smoothly c

was constructed with partitions of unity subordinate to an atlaof positively oriented charts with an integer set H

M (n, m ) andI }a partition of unity subordinate to C

M . Since suppω is c

Ddω =

α∈I Dd(gα ω),


ω =n

h=1

(−1)h− 1ωμ i h ν i hdxμ i 1 ν i 1

∧ · · ·∧

dxμ i h ν i h∧ · · ·d

where dxμ i h ν i h means that dxμ i h ν i h is deleted, where

ih∈ {1, · · ·, nU , (1( nU + 1)) , · · ·, (1n1), (2( nU + 1)) , · · ·, (2n2)

Then

dω =n

h=1

∂ωμ i h ν i h

∂x μ i h ν i hdxμ i 1 ν i 1

∧ · · ·∧dxμ i n ν i n . (5 −Consider the appearance of neighborhood U . There are tw

considered.

Case 1 U ∂ D = ∅In this case ω = 0 and U is in M \ D or in Int D The f



In this case, ∂ D ω = 0 and U is in M \ D or in Int D. The fimplies that D d(gα ω) = 0. For the later, we nd that

D dω =

n

h=1 U

∂ωμ i h ν i h

∂x μ i h ν i h dxμ i

1ν i

1 · · ·dxμ i

nν i

n . (5 −Notice that + ∞

−∞∂ω μ i h

ν i h

∂x μ i hν i h

dxμ i h ν i h = 0 since ωμ i h ν i hhas compac

D dω = 0 as desired.

Case 2 U ∂ D = ∅


∂ Dω = U ∩∂ D

ω

= n

h=1

(−1)h− 1 U ∩∂ Dωμ i h ν i h

dxμ i 1 ν i 1∧ · · ·∧

dxμ i

= ( −1)n − 1 U ∩∂ Dωμ i n ν i n

dxμ i 1 ν i 1∧ · · ·∧dxμ i n − 1 ν

since dxμ i n ν i n (q) = 0 for q ∈U ∩∂ D . Notice that R n− 1

orientation on R n − 1 is not the boundary orientation, whose o

−en = (0 , · · ·, 0, −1). Hence

∂ Dω = − ∂ R n

+

ωμ i n ν i n(xμ i 1 ν i 1 , · · ·, xμ i n − 1 ν i n − 1 , 0)dxμ i 1 ν

On the other hand, by the fundamental theorem of calcu



R n − 1( ∞

0

∂ωμ i n ν i n

∂x μ i n ν i n)dxμ i 1 ν i 1 · · ·dxμ i n − 1 ν i n − 1

=

− R n − 1

ωμ i n ν i n(xμ i 1 ν i 1 ,

· · ·, xμ i n − 1 ν i n − 1 , 0)dxμ i 1 ν

Since ωμ i n ν i nhas a compact support, thus

U ω = − R n − 1

ωμ i n ν i n(xμ i 1 ν i 1 , · · ·, xμ i n − 1 ν i n − 1 , 0)dxμ i 1 ν


Ddω = ∂ D

ω,

particularly, if M is nothing but a manifold, the Stokes’ theorem

Corollary 5.7.2 Let M be a smoothly combinatorial manifold wH

M (n, m ). For n∈H

M (n, m ), if ω∈Λn (M ) has a compact supp

M ω = 0 .

By the denition of integration on vertex-edge labeled graphslet a boundary subset of G([0, nm ], [0, nm ]) mean that of its correnatorial manifold M . Theorem 5.7.1 and Corollary 5.7.2 then cancombinatorial manner as follows.

Theorem 5.7.2 Let G([0, nm ], [0, nm ]) be a vertex-edge labeled grset H G (n, m ) and D a boundary subset of G([0, nm ], [0, nm ]). For ω∈Λn (G([0, nm ], [0, nm ])) has a compact support, then



Ddω = ∂ D

ω

with the convention

∂ D

ω = 0 , while ∂ D =

∅

.

Corollary 5.7.3 Let G([0, n m ], [0, n m ]) be a vertex-edge labeled grset H G (n, m ). For ∀n ∈

H G (n, m ) if ω ∈Λn (G([0, nm ], [0, nm

support, then

ω = 0 .


dω = (∂A∂x 1 −

∂B∂x 2

)dx1∧dx2.

Whence, the Green’s formula is nothing but a special case o

Ddω = ∂ D

ω

with D = D.

Example 5.7.2 Let S be a surface in R 3 with boundary su

simple curve with a direction. We have know the classical S

∂S Adx1 + Bdx 2 + Cdx3

= S (

∂C ∂x 2 −

∂B∂x 3

)dx2dx3 + (∂A∂x 3 −

∂C ∂x 1

)dx3dx1 + (

Now let ω = Adx1 + Bdx 2 + Cdx3∈Λ10(R 3). Then we

∂C ∂B ∂A ∂C ∂



dω = (∂C ∂x 2 −

∂B∂x 3

)dx2∧dx3 + (∂A∂x 3 −

∂C ∂x 1

)dx3∧dx1 + (∂∂

Whence, the classical Stokes’ formula is a special case of th

Ddω = ∂ D

ω

in Theorem 5.7.1 with D = S .

5.7.2 Combinatorial Gauss’ Theorem. Let D be a domary and a positive direction determined by its normal vector


By denition, we know that the Lie derivative forms a Lie a

Theorem 5.7.3 The Lie derivative LX Y = [X, Y ] on X (M ) foi.e.,

(i) [ , ] is R -bilinear;

(ii ) [X, X ] = 0 for all X ∈X (M );

(iii ) [X, [Y, Z ]] + [Y, [Z, X ]] + [Z, [X, Y ]] = 0 for all X,Y,Z

Proof These brackets [ X, Y ] forms a Lie algebra can be imm

Theorem 5 .1.2 and its denition.Now we nd the local expression for [X, Y ]. For p∈M , let (U

U p →R (n1( p), · · ·, n s ( p)( p)) be a local chart of p and X, Y the locof X, Y . According to Theorem 5 .7.3, the local representative oWhence,

[X, Y ][f ](x) = X [Y [f ]](x) −Y [X [f ]](x)



= D(Y [ f ])(x) ·X (x) −D(X [ f ])(x) ·for f ∈

X p(M ). Now Y [

f ](x) = D

f (x) ·Y (x) and maybe calcul

ruler. Notice that the terms involving the second derivative of symmetry of D 2

f (x). We are left with

D f (x) ·(DY (x) ·X (x) −DX (x) ·Y (x)),

which implies that the local representative of [ X, Y ] is DY ·X −


Denition 5.7.3 For X 1, · · ·, X k∈X (M ), ω∈Λk+1 (M ),

iX ω(X 1, · · ·, X k) = ω(X, X 1, · · ·, X k)

Then we have the following result.

Theorem 5.7.4 For integers k, l ≥0, if ω∈Λk(M ), ∈Λ

(i) iX (ω∧ ) = ( iX ω)∧ + ( −1)kω∧iX ;

(ii ) LX ω = iX dω + diX ω.

Proof By denition, we know that iX ω

∈

Λk− 1()M . Fo

iX (ω∧ )(u2, · · ·, uk+ l) = ω∧ (u1, u2, · ·and

(iX ω)

∧

+ (

−1)kω

∧

iX =(k + l −1)!

(k −1)!l!A (iX

+( −1)k k + l −1k!(l −1)



by Denition 5.2.2. Let

σ0 =2 3

· · ·k + 1 1 k + 2

· · ·1 2 · · · k k + 1 k + 2 · · ·Then we know that each permutation in the summation of written as σσ0 with signσ0 = ( −1)k . Whence,

k (k + l 1)! (k + l 1)!


LX f = df = iX df.

Now assume it holds for an integer l. Then a ( l + 1)-form mdf ∧ω. Notice that LX (df ∧ω) = LX df ∧ω + df ∧LX ω since wa tensor derivation by denition. Applying ( i), we know that

iX d(df ∧ω) + diX (df ∧ω) = −iX (df ∧dω) + d(iX df ∧ω

= −iX df ∧dω) + df ∧iX dω+ diX df ∧ω + iX df ∧ω + df

= df ∧LX ω + dLX f ∧ω

by the induction assumption. Notice that dLX f = LX df , we get

Denition 5.7.4 A volume form on a smoothly combinatorial manω in Λn for some integers n∈

H M (n, m ) such that ω( p) = 0 fo

is a vector eld on M , the unique function divωX determined by



is a vector eld on M , the unique function divωX determined by called the divergence of X and incompressible if divωX = 0 .

Then we know the generalized Gauss’ theorem on smoothly co

ifolds following.

Theorem 5.7.5 Let M be a smoothly combinatorial manifold wH

M (n, m ), D a boundary subset of M and X a vector eld on Msupport. Then

Sec.5.8 Combinatorial Finsler Gometry

D(divX )v = ∂ D

iX v .

by Theorem 5 .7.1.

Then the Gauss’ theorem in R 3 is generalized on smootifolds in the following.

Theorem 5.7.6 Let (M, g) be a homogenously combinatoriacarrying a outward-pointing unit normal n ∂ M along ∂ M a

(M, g) with a compact support. Then

M (divX )dvM = ∂ M

X, n ∂ M dv∂ M

where v and v∂ M are volume form on M , i.e., nonzero elemX, n ∂ M the inner product of matrixes X and n ∂ M .

Proof Let v∂ M be the volume element on ∂ M induced ume element vM ∈Λn (M )(M ), i.e., for any positively orientedT (∂M ) we have that



T p(∂ M ), we have that

v∂ M (x)(v1, · · ·, vn (M )− 1) = vM (−∂

∂x n (M ), v1, · ·

Now since

(iX vM )(x)(v1, · · ·, vn (M )− 1) = vM (x)(X i(x)v i −X n (M )(x)∂

= X n (M )(x)v∂M (x)(v1, , vn


§5.8 COMBINATORIAL FINSLER GEOMETRY

5.8.1 Combinatorial Minkowskian Norm. A Minkowskianspace V is dened in the following denition, which can be asmoothly combinatorial manifolds.

Denition 5.8.1 A Minkowskian norm on a vector space V is a fusuch that

(1) F is smooth on V \{0}and F (v) ≥0 for ∀v∈V ;

(2) F is 1-homogenous, i.e., F (λv) = λF (v) for ∀λ > 0;(3) for all y∈V \{0}, the symmetric bilinear form gy : V ×

gy(u, v) =i,j

∂ 2F (y)∂y i∂y j

is positive denite for u, v∈V .

Denoted by T M = p∈M

T pM .

5 8 2 Combinatorial Finsler Geometry A combinatorial F



5.8.2 Combinatorial Finsler Geometry. A combinatorial Fon a Minkowskian norm is dened on T M following.

Denition 5.8.2 A combinatorial Finsler geometry is a smoothmanifold M endowed with a Minkowskian norm F on T M , deno

Then we get the following result.

Theorem 5.8.1 There are combinatorial Finsler geometries.

Proof Let M (n1, n2, · · ·, nm ) be a smoothly combinatorial m

Sec.5.8 Combinatorial Finsler Gometry

M ∈V (GL [M (n 1 ,n 2 ,··· ,n m )]){(U Mα ;ϕMα )|α∈I

By the decomposition theorem for unit, we know that there hMα , α∈I M such that

M ∈V (GL [M (n 1 ,n 2 ,··· ,n m )]) α∈I M

hMα = 1 with 0 ≤Now we choose a Minkowskian norm F Mα on T pM α for

F Mα =hMα F Mα , if p∈U Mα ,

0, if p∈U Mα

for∀ p∈M . Now let

F = M ∈V (GL [M (n 1 ,n 2 ,··· ,n m )]) α∈I F Mα .

Then F is a Minkowskian norm on T M (n1, n2, · · ·, nm ) si di l h ll di i (1) (3) i D i i 5 8 1



immediately that all conditions (1) −(3) in Denition 5.8.1

5.8.3 Geometrical Inclusion. For the relation of combi

tries with these Smarandache multi-spaces, we obtain the nex

Theorem 5.8.2 A combinatorial Finsler geometry (M (n1, n 2

dache geometry if m ≥2.

Proof Notice that if m ≥ 2, then M (n1, n2, · · ·, nm ) itwo manifolds M n 1 and M n 2 with n1 = n2. By denition, w


Corollary 5.8.1 There are inclusions among Smarandache multgeometry, Riemannian geometry and Weyl geometry :

{Smarandache geometries }⊃ {combinatorial Finsler ge

⊃ {Finsler geometry }and {combinatorial Riemannian g

⊃ {Riemannian geometry }⊃ {Weyl geometry }.

Proof Let m = 1. Then a combinatorial Finsler geometry ( M

is nothing but just a Finsler geometry. Applying Theorems 5 .8.special case, we get these inclusions as expected.

Corollary 5.8.2 There are inclusions among Smarandache geomerial Riemannian geometries and K¨ ahler geometry :

{Smarandache geometries } ⊃ {combinatorial Riemannian

⊃ {Riemannian geometry }{K ¨hl g t }



⊃ {K ahler geometry }.

Proof Let m = 1 in a combinatorial manifold M (n1, n2, · ·Theorems 5 .3.4 and 5.8.2, we get inclusions

{Smarandache geometries }⊃ {combinatorial Riemannian g

⊃ {Riemannian geometry }.For the K ahler geometry, notice that any complex manifold

smoothly real manifold M 2n with a natural base {∂ ∂x i , ∂

∂y i }for T pn




Of course, nearly all existent operators with local properties oor Riemannian geometries can be reconstructed in these combinaRiemannian geometries and nd the local forms similar to those i

mannian geometries.5.9.4 Global properties To nd global properties on manifoldin classical differential geometry. The same is true for combinIn classical geometry on manifolds, some global results, such as theorem and Atiyah-Singer index theorem,..., etc. are well-known

the pth

de Rham cohomology group on a manifold M and theFredholm operator D: H k (M, E ) →L2(M, F ) are dened to be a

H p(M ) =Ker (d : Λ p(M ) →Λ p+1 (M ))Im (d : Λ p− 1(M ) →Λ p(M ))

.

and an integer

IndD= dim Ker (D) −dim(L2(M, F )

ImD)

i l Th d Rh h d h A i h Si i d



respectively. The de Rham theorem and the Atiyah-Singer index tively conclude that

for any manifold M , a mapping ϕ : Λ p(M ) →Hom (Π p(natural isomorphism ϕ∗ : H p(M ) →H n (M ; R ) of cohomology grois the free Abelian group generated by the set of all p-simplexes in

and

Sec.5.9 Remarks

5.9.5 Combinatorial Gauss-Bonnet Theorem. We Bonnet formula in the nal section of Chapter 3. Thenin combinatorial differential geometry? Particularly, wether

Gauss-Binnet-Chern result

M 2pΩ = χ (M 2 p)

for an oriently compact Riemannian manifold (M 2 p, g), whe

Ω =(

−1) p

22 pπ p p!i1 ,i 2 ,··· ,i 2p

δi1 ,··· ,i 2p

1,··· ,2 p Ωi1 i2

∧ · · ·∧Ωi

and Ωij is the curvature form under the natural chart {ei}o

δi1 ,··· ,i 2p1,··· ,2 p =⎧⎪

⎪⎨⎪⎪⎩

1, if permutation i1 · · ·i2 p is

−1, if permutation i1 · · ·i2 p i

0, otherwise .

to combinatorial Riemannian manifolds (M,g, D) such that



M 2nΩ = χ (M 2n )

with

Ω =(−1)n

22n πn n!i1 ,i 2 ,··· ,i 2n

δi1 ,··· ,i 2n1,··· ,2n Ω(i1 j 1 )( μ2 ν 2 )∧ · · ·∧Ω(i2n

CHAPTER 6.

Combinatorial Riemannian Submanifolds

Principal Fibre Bundles

For the limitation of human beings, one can only observes pWORLD. Even so, the Whitney’s result asserted that one can recowhole WORLD in a Euclidean space. The same thing also happennatorial manifolds, i.e., how do we realize multi-spaces or combinifolds? how do we apply them to physics? This chapter presentsanswers for the two questions in mathematics. Analogous to the clasetry, these Gauss ’s, Codazzi ’s and Ricci ’s formulae or fundament



are established for combinatorial Riemannian submanifolds SectionSection 6.3 considers the embedded problem of combinatorial ma

shows that any combinatorial Riemannian manifold can be isometbedded into combinatorial Euclidean spaces. Section 6 .4 generaltopological or Lie groups to topological or Lie multi-groups, whicapplications of combinatorial manifolds. This section also considebras of Lie multi-groups. Different from the classical case, we es

Sec.6.1 combinatorial Riemannian Submanifolds

§6.1 COMBINATORIAL RIEMANNIAN SUBMANIFOLD

6.1.1 Fundamental Formulae of Submanifold. We haically combinatorial submanifolds in Section 4 .2, i.e., a comor combinatorial combinatorial Riemannian submanifold S ismanifold or a combinatorial Riemannian manifold M such thtorial manifold or a combinatorial Riemannian manifold. In section, we generalize conditions on differentiable submanithe Gauss ’s, the Codazzi ’s and the Ricci ’s formulae or fun

handling the behavior of submanifolds of a Riemannian manRiemannian manifolds.

Let (i, M ) be a smoothly combinatorial submanifold of a(N, gN , D). For ∀ p∈M , we can directly decompose the taninto

T pN = T pM ⊕T ⊥ p M

on the Riemannian metric gN at the point p, i.e., choice



T ⊥ p M to be gN |T p M or gN |T ⊥p M , respectively. Then we get

T pM and a orthogonal complement T ⊥ p M of T pM in T pN , i

T ⊥ p M = {v∈T pN | v, u = 0 for ∀u∈T p

We call T pM , T ⊥ p M the tangent space and normal space of (N, gN , D), respectively. They both have the Riemannian strucis a combinatorial Riemannian manifold under the induced m

284 Chap.6 combinatorial Riemannian Submanifolds with Principal F

Whence, for ∀X, Y ∈X (M ), we know that

DX Y = DX Y + D⊥X Y,

called the Gauss formula on the combinatorial Riemannian subwhere DX Y = ( DX Y ) and D⊥

X Y = ( DX Y )⊥.

Theorem 6.1.1 Let (i, M ) be a combinatorial Riemannian submaniwith an induced metric g = i∗gN . Then for ∀X,Y,Z , D : X

X (M ) determined by D (Y, X ) = DX Y is a combinatorial Riemaon (M, g) and D⊥ : X (M )

×X (M )

→T ⊥(M ) is a symmetr

tensor eld of order 2, i.e.,

(1) D⊥X + Y Z = D⊥

X Z + D⊥Y Z ;

(2) D⊥λX Y = λD⊥

X Y for ∀λ∈C ∞ (M );

(3) D⊥X Y = D⊥

Y X .

Proof By denition, there exists an inclusion mapping i : M(i, M ) is a combinatorial Riemannian submanifold of ( N, gN , D) i∗gN .



For ∀X,Y,Z ∈X (M ), we know that

DX + Y Z = DX Z + DY Z

= ( DX Z + DX Z ) + ( D⊥

X Z + D⊥

X Z )

by properties of the combinatorial Riemannian connection D. The

⊥ ⊥ ⊥


and

D⊥X (λY ) = λD⊥

X Y.

Thereafter, the mapping D : X (M )

×X (M )

→X (M )

nection on (M, g) and D⊥ : X (M ) ×X (M ) →T ⊥(M ) h(2).

By the torsion-free of the Riemannian connection D, i.

DX Y

−DY X = [X, Y ]

∈

X (M )

for∀X, Y ∈X (M ), we get that

DX Y −DY X = ( DX Y −DY X ) = [X,

and

D⊥

X Y −D⊥

Y X = ( DX Y −DY X )⊥ = 0

i.e., D⊥X Y = D⊥

Y X . Whence, D is also torsion-free on (M,



, X Y , ( ,on D⊥ holds. Applying the compatibility of D with gN in (Nthat

Z X, Y = DZ X, Y + X, DZ Y

= DZ X, Y + X, DZ Y


Theorem 6.1.2 Let (i, M ) be a combinatorial Riemannian submaniwith an induced metric g = i∗gN . Then the mapping D⊥ : T ⊥M determined by D(Y ⊥, X ) = D⊥

X Y ⊥ is a combinatorial Riemannia

T ⊥

M .Proof By denition, we have known that there is an inclusion

N such that ( i, M ) is a combinatorial Riemannian submanifold oa metric g = i∗gN . For ∀X, Y ∈

X (M ) and ∀Z ⊥, Z ⊥1 , Z ⊥2 ∈T ⊥M

D⊥

X + Y Z ⊥

= D⊥

X Z ⊥

+ D⊥

Y Z ⊥

, D⊥

X (Z ⊥

1 + Z ⊥

2 ) = D⊥

X Z ⊥

1

similar to the proof of Theorem 6 .1.4. For ∀λ∈C ∞ (M ), we know

DλX Z ⊥ = λDX Z ⊥, DX (λZ ⊥) = X (λ)Z ⊥ + λDX

Whence, we nd that

D⊥

λX Z ⊥ = ( λDX Z ⊥)⊥ = λ(DX Z ⊥)⊥ = λD⊥

X Z ⊥



D⊥

X (λZ ⊥) = X (λ)Z ⊥ + λ(DX Z ⊥)⊥ = X (λ)Z ⊥ + λD

Therefore, the mapping D⊥

: T ⊥

M ×X

(M ) →T ⊥

M is a combinon T ⊥M . Applying the compatibility of D with gN in (N, gN ,that

X Z ⊥1 , Z ⊥2 = DX Z ⊥1 , Z ⊥2 + Z ⊥1 , DX Z ⊥2 = D⊥X Z ⊥1 , Z ⊥2 +

which implies that D⊥ : X (M ) X (M ) X (M ) is a combina


X (M ) is a tensor eld of type (1, 1). Besides, if DZ ⊥ islinear transformation on M , then DZ ⊥ : T pM →T pM at self-conjugate transformation on g, i.e., the equality followin

DZ ⊥ (X ), Y = D⊥X (Y ), Z ⊥ , ∀X, Y ∈T p

Proof First, we establish the equality (6 −1). By applyingDX Z ⊥, Y + Z ⊥, DX Y and Z ⊥, Y = 0 for ∀X, Y ∈

X

we nd that

DZ ⊥ (X ), Y = DX Z ⊥, Y

= X Z ⊥, Y − Z ⊥, DX Y =

Thereafter, the equality (6 −1) holds.Now according to Theorem 6 .1.1, D⊥

X Y posses tensorT pM . Combining this fact with the equality (6 −1), DZ ⊥ (X )(1, 1). Whence, DZ ⊥ determines a linear transformation DZ

point p∈M . Besides, we can also show that DZ ⊥ (X ) poss⊥ ⊥



for∀Z ⊥∈T ⊥M . For example, for any λ∈C ∞ (M ) we know

DλZ ⊥ (X ), Y = D⊥

X Y,λZ ⊥ = λ D⊥

X Y,

= λDZ ⊥ (X ), Y , ∀X, Y

by applying the equality (6 −1) again. Therefore, we nallλDZ ⊥ (X ).


6.1.2 Local Form of Fundamental Formula. Now we lofor D and D⊥. Let (M,g, D ) be a combinatorial Riemannia(N, gN , D). For ∀ p∈M , let

{eAB |1 ≤A ≤dN ( p), 1 ≤B ≤nA and eA1 B = eA2 B ,

for 1 ≤A1, A2 ≤dN ( p) if

be an orthogonal frame with a dual

{ωAB |1 ≤A ≤dN ( p), 1 ≤B ≤nA and ωA1B = ωA2 B ,

for 1 ≤A1, A2 ≤dN ( p) i

at the point p in T N abbreviated to {eAB }and ωAB . Choose index(ab), (cd),

· · ·and (αβ ), (γδ ),

· · ·satisfying 1

≤A, C

≤d

N ( p)

1 ≤D ≤nC , · · ·, 1 ≤a, c ≤dM ( p), 1 ≤b ≤na , 1 ≤d ≤nc, · · ·andor β, δ ≥n i + 1 for 1 ≤ i ≤dM ( p). For getting local forms of Deven assume that {eAB }, {eab}and {eαβ }are the orthogonal fram



{ }{ } { }the tangent vector space T N, T M and the normal vector space T3.1

−3.3. Then the Gauss’s and Weingarten’s formula can be expr

D eab ecd = D eabecd + D⊥

eabecd ,

Deab eαβ = Deabeαβ + D⊥

eabeαβ .

When p is varied in M we know that ωab = i∗(ωab) and ωαb = 0

Sec.6.2 Fundamental Equations on Combinatorial Submanifolds

De AB = ωCDAB eCD

by denition. We nally get that

De ab = ωcdab ecd + ωαβ

ab eαβ , De αβ = ωcdαβ ecd +

Since dωαi = ωab∧ωαi

ab = 0 , dωiβ = ωab∧ωiβ

ab = 0, by theTheorem 5 .2.3, we know that

ωαi

ab= hαi

(ab)( cd)ωcd, ωiβ

ab= h iβ

(ab)( cd)ωcd

with hαi(ab)( cd) = hαi

(cd)( ab) and hiβ (ab)( cd) = hiβ

(cd)( ab) . Thereafter, w

D⊥

eabecd = ωαβ

cd (eab)eαβ = hαβ (ab)( cd) eαβ ,

D eab eαβ = ωcdαβ (eab)ecd = hαβ (ab)( cd)eαβ .

Whence, we get local forms of D and D⊥ in the following.

Theorem 6.1.4 Let (M,g, D ) be a combinatorial Riem



(N, gN , D). Then for ∀ p∈M with locally orthogonal framesdual

{ωAB

},

{ωab

}in T N , T M ,

D eabecd = ωcd

αβ (eab)ecd , D⊥

eabecd = hαβ

(ab)( cd

Deabeαβ = hαβ

(ab)( cd) eαβ , D⊥

eabeαβ = ωγδ

αβ (eab


§6.2 FUNDAMENTAL EQUATIONS ON

COMBINATORIAL SUBMANIFOLDS

6.2.1 Gauss Equation. Applications of these Gauss’s andmulae enable one to get fundamental equations such as the GausRicci ’s equations on curvature tensors for characterizing combinatosubmanifolds.

Theorem 6.2.1(Gauss equation) Let (M,g, D ) be a combinat

submanifold of (N, gN , D) with the induced metric g = i∗gN andtensors on M and N , respectively. Then for ∀X,Y,Z,W ∈

X (M

R(X,Y,Z,W ) = RN (X,Y,Z,W ) + D⊥

X Z, D⊥

Y W − D⊥

Y


RN (X, Y )Z = DX DY Z −DY DX Z −D [X,Y ]Z.



Applying the Gauss formula, we nd that

RN (X, Y )Z = DX (DY Z + D⊥

Y Z ) −DY (DX Z + D⊥

X

−(D [X,Y ]Z + D⊥

[X,Y ]Z )

= DX DY Z + D⊥

X DY Z + DX D⊥

Y Z −DY

D⊥D Z D D⊥Z D Z D

Sec.6.2 Fundamental Equations on Combinatorial Submanifolds

R(X, Y )Z, W = RN (X, Y )Z, W + D⊥

X Z, D⊥

Y W

by applying the equality (6 −1) in Theorem 6.1.3, i.e.,

R(X,Y,Z,W ) = RN (X,Y,Z,W ) + D⊥

X Z, D⊥

Y W −

6.2.2 Codazzi Equation. For

∀

X,Y,Z

∈

X (M ), dene tDX on D⊥

Y Z by

(DX D⊥)Y Z = D⊥

X (D⊥

Y Z ) −D⊥

D X Y Z −D⊥

Y (

Then we get the Codazzi equation in the following.

Theorem 6.2.2 (Codazzi equation) Let (M,g, D ) be a comsubmanifold of (N, gN , D) with the induced metric g = i∗gtensors on M and N , respectively. Then for ∀X,Y,Z ∈

X (



(DX D⊥)Y Z −(DY D⊥)X Z = R⊥(X, Y

Proof Decompose the curvature tensor RN (X, Y )Z int

RN (X, Y )Z = RN (X, Y )Z + R⊥N (X, Y )

Notice that


6.2.3 Ricci Equation. For ∀X, Y ∈X (M ), Z ⊥∈T ⊥(M ), the

R⊥ determined by D⊥ in T ⊥M is dened by

R⊥(X, Y )Z ⊥ = D⊥

XD⊥

Y Z ⊥

−D⊥

Y D⊥

XZ ⊥

−D⊥

[X,Y ]Z

Similarly, we get the next result.

Theorem 6.2.3(Ricci equation) Let (M,g, D ) be a combinatsubmanifold of (N, gN , D) with the induced metric g = i∗gN andtensors on M and N , respectively. Then for ∀X, Y ∈

X (M ), Z ⊥

R⊥(X, Y )Z ⊥ = R⊥N (X, Y )Z ⊥ + ( DX D⊥)Y Z ⊥−(DY D⊥

Proof Similar to the proof of Theorem 6 .2.1, we know that

RN

(X, Y )Z ⊥ = DX DY Z ⊥

−DY DX Z ⊥

−D [X,Y ]Z ⊥

= R⊥(X, Y )Z ⊥ + D⊥

X DY Z ⊥−D⊥

Y DX Z

+ DX D⊥

Y Z ⊥−DY D⊥

X Z ⊥

= ( R⊥(X, Y )Z ⊥ + ( DX D⊥)Y Z ⊥−(DY D



+ DX D⊥

Y Z ⊥−DY D⊥

X Z ⊥.

Whence, we get that

R⊥(X, Y )Z ⊥ = R⊥N (X, Y )Z ⊥ + ( DX D⊥)Y Z ⊥−(DY D⊥

6 2 4 Local Form of Fundamental Equation We can also



294Chap.6 combinatorial Riemannian Submanifolds with Principal F

(ΩN )αβ ab = dωαβ

ab −ωcdab∧ωαβ

cd −ωγδab ∧ωαβ

γδ

= d(hαβ (ab)( cd) ω

cd)

−hαβ

(cd)( ef )ωcdab

∧

ωef

−hγδ

(ab)( ef )ωef

= ( dhαβ (ab)( cd) −hαβ

(ab)( ef ) ωef cd ) −hαβ

(ef )( cd) ωef ab + hγδ

(ab)( c

= hαβ (ab)( cd)( ef ) ω

ef ∧ωcd

=12

(hαβ (ab)( cd)( ef ) −hαβ

(ab)( ef )( cd) )ωef ∧ωcd

and

(ΩN )γδαβ = dωγδ

αβ −ωef αβ ∧ωγδ

ef −ωζηαβ ∧ωγδ

ζη

= Ω⊥γδαβ +

12

e,f

(hαβ (ef )( ab) h

γδ(ef )( cd) −hαβ

(ef )( cd) hγδ(ef )( ab

These equalities enables us to get

hαβ (ab)( cd)( ef ) −hαβ

(ab)( ef )( cd) = ( RN )(αβ )( ab)( cd)( ef ) ,

and



R⊥(αβ )( γδ )( ab)( cd) = ( RN )(αβ )( γδ )( ab)( cd) − e,f (hαβ (ab)( ef ) hγδ(cd)( gh ) −hα(

These are just the Codazzi ’s or Ricci ’s equations.

Sec.6.3 Embedded Combinatorial Submanifolds

(F ∗ω)(v) = ω(F ∗(v)) for ∀ω∈T ∗F ( p)N and ∀v

is called a pull-back mapping. We know that mappings F ∗ a

For a smooth mapping F : M →N and p∈M , if F ∗ p :to-one, we call it an immersion mapping . Besides, if F ∗ p is onis a homoeomorphism with the relative topology of N , then mapping and (F, M ) a combinatorial embedded submanifold . inclusion mapping i : M →N and denoted by ( i, M ) a comof N .

Now let M = M (n1, n 2, · · ·, nm ), N = N (k1, k2, · · ·, kl)natorial manifolds and F : M →N a smooth mapping. Foand (V F ( p) , ψF ( p)) be local charts of p in M and F ( p) in N , r

J X ;Y (F )( p) = [∂F κλ

∂x μν ]

the Jacobi matrix of F at p. Then we nd that

Theorem 6.3.1 Let F : M →N be a smooth mapping froan immersion mapping if and only if



rank (J X ;Y (F )( p)) = dM

( p)

for ∀ p∈M .

Proof Assume the coordinate matrixes of points p∈[xij ]s ( p)× n s ( p ) and [yij ]s(F ( p)) × n s ( F ( p )) , respectively. Notice that

296Chap.6 combinatorial Riemannian Submanifolds with Principal F

F ∗ p(∂

∂x ij )(f ) =∂ (f ◦F )

∂x ij

= μ,ν

∂F μν

∂x ij

∂f ∂yμν .

Whence, we nd that

F ∗ p(∂

∂x ij ) =μ,ν

∂F μν

∂x ij∂

∂yμν .

According to a fundamental result on linear equation systems,tions in the equation system (6 −3) if and only if

rank (J X ;Y (F )( p)) = rank (J ∗X ;Y (F )( p)) ,

where

⎡⎢⎢⎢

· · · F ∗ p( ∂ ∂x 11 )

· · · · · ·F ( ∂ )

⎤



J ∗X ;Y (F )( p) =⎢⎢⎢

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

· · · F ∗ p( ∂ ∂x 1n 1 )

J X ;Y (F )( p)

· · ·· · · F ∗ p( ∂ ∂x s ( p )1 )

· · · · · ·· · · F ∗ p( ∂

∂x s ( p ) n s ( p ))⎦

.

We have known that


Theorem 6.3.2 Let M be a smoothly combinatorial manifIf for ∀M ∈V (GL [M ]), there exists an embedding F M : Membedded into N .

Proof By assumption, there exists an embedding F MV (GL [M ]). For p∈M , let V p be the intersection of s( p) maniwith functions f M i , 1 ≤ i ≤ s( p) in Lemma 2.1 existed. DeN at p by

F ( p) = s( p)

i=1f M i F M i .

Then F is smooth at each point in M for the smooth of eachT pN is one-to-one since each (F M i )∗ p is one-to-one at the pobe embedded into the manifold N .

Theorem 6.3.3 Let M and N be smoothly combinatorial mV (GL [M ]), there exists an embedding F M : M →N , then MN .

Proof Applying Theorem 5 .6.2, we can get a mapping



Proof Applying Theorem 5 .6.2, we can get a mapping

F ( p) = s( p)

i=1

f M i F M i

at ∀ p∈M . Similar to the proof of Theorem 2.2, we knowF ∗ p : T pM →T pN is one-to-one. Whence, M can be embedd


Given two topological spaces C 1 and C 2, a topological embeda one-to-one continuous map

f : C 1

→C 2.

When f : M (n1, n2, · · ·, nm ) →R (k1, · · ·, kl) maps each manifolclidean space of R (k1, · · ·, kl), we say that M is in-embedded into

Whitney had proved once that any n-manifold can be topoloa closed submanifold of R 2n +1 with a sharply minimum dimensio

([AbM1]) . Applying Whitney’s result enables us to nd conditionsbinatorial manifold embedded into a combinatorial Euclidean space

Theorem 6.3.4 Any nitely combinatorial manifold M (n1, n 2, ·bedded into R 2n m +1 .

Proof According to Whitney’s result, each manifold M n i

M (n1, n2, · · ·, nm ) can be topological embedded into a Euclidean η ≥ 2n i + 1. By assumption, n1 < n 2 < · · ·< n m . WhenceM (n1, n2, · · ·, nm ) can be embedded into R 2n m +1 . Applying Theorthat M (n1, n 2, · · ·, n m ) can be embedded into R 2n m +1 .



For in-embedding a nitely combinatorial manifold M (n1, n 2,

binatorial Euclidean spaces R (k1, · · ·, kl), we get the next result.

Theorem 6.3.5 Any nitely combinatorial manifold M (n1, n 2, ·embedded into a combinatorial Euclidean space R (k1, · · ·, kl) if th

: n1 n 2 nm k1 k2 kl




For two given combinatorial Riemannian C r -manifolds (M,g, DM

an isometry embedding

i : M

→N

is an embedding with g = i∗gN . By those discussions in Sections 6local charts of M , N be (U, [x]), (V, [y]) and the metrics in M , N

gN =(ςτ ),(ϑι )

gN ( ςτ )( ϑι )dyςτ ⊗dyϑι , g =

(μν ),(κλ )

g(μν )( κλ )dxμ

then an isometry embedding i form M to N need us to determinefunctions

yκλ = iκλ [xμν ], 1 ≤μ ≤s( p), 1 ≤ν ≤ns ( p)

for

∀

p

∈

M such that

R (ab)( cd)( ef )( gh ) = ( RN )(ab)( cd)( ef )( gh ) −α,β

(hαβ (ab)( ef )h

αβ (cd)( gh ) −hα

(



hαβ (ab)( cd)( ef )

−hαβ

(ab)( ef )( cd) = ( RN )(αβ )( ab)( cd)( ef ) ,

R⊥(αβ )( γδ )( ab)( cd) = ( RN )(αβ )( γδ )( ab)( cd) −e,f

(hαβ (ab)( ef ) h

γδ(cd)( gh ) −hα

(

with R⊥(αβ )( γδ )( ab)( cd) = R(eab , ecd)eαβ , eγδ ,


(ςτ )

∂i ςτ

∂yμν ∂i ςτ

∂yκλ = g(μν )( κλ ) [y]

since a combinatorial Euclidean space R (k1,

· · ·, kl) is equ

space R k with a constant k = l( p) +l( p)

i=1(ki − l( p)) for ∀ p∈

on p (see [9] for details) and the metric of a Euclidean space

gR =μ,ν

dyμν ⊗dyμν .

These combined with additional conditions enable us to nd nconditions for existing particular combinatorial Riemannian s

Similar to Theorems 6 .3.4 and 6.3.5, we can also get isometry embedding by applying Theorem 5 .6.2, i.e., the parwe need two important lemmas following.

Lemma 6.3.1([ChL1]) For any integer n ≥ 1, a Riemannmensional n with 2 < r ≤ ∞can be isometrically embedded iR n 2 +10 n +3 .

Lemma 6.3.2 Let (M,g, DM ) and (N, gN , D) be combinat



( g M ) ( gN )ifolds. If for

∀

M

∈

V (GL [M ]), there exists an isometry embthen M can be isometrically embedded into N .

Proof Similar to the proof of Theorems 6 .3.2 and 6.3.3that the mapping F : M →N dened by

s( p)


= s ( p)

i=1 s( p)

j =1

gN (f M i (F M i )(v), f M j (F M

= s ( p)

i=1 s( p)

j =1

g(f M i (F M i )(v), f M j (F M j )

= g( s( p)

i=1

f M i v, s( p)

j =1

f M j w)

= g(v, w).

Therefore, F is an isometry embedding.Applying Lemmas 6.3.1 and 6.3.2, we get results on isometr

combinatorial manifolds into combinatorial Euclidean spaces follow

Theorem 6.3.6 Any combinatorial Riemannian manifold M (n1,isometrically embedded into R n 2

m +10 n m +3 .

Proof According to Lemma 6.3.1, each manifold M n i , 1 ≤ i

· · ·, n m ) can be isometrically embedded into a Euclidean spacen2

i + 10 n i + 3. By assumption, n1 < n 2 < · · ·< n m . Thereafter,M (n1, n2, · · ·, nm ) can be embedded into R n 2

m +10 n m +3 . Applyingknow that M (n n n ) can be isometrically embedded into



know that M (n1, n 2, · · ·, n m ) can be isometrically embedded into

Theorem 6.3.7 A combinatorial Riemannian manifold M (n1, nisometrically embedded into a combinatorial Euclidean space R (is an injection

: {n1, n 2, · · ·, nm} → {k1, k2, · · ·, kl}

Sec.6.4 Topological Multi-Groups

Proof If

(n i) ≥max{2 + 10 + 3 | ∀∈ − 1( (n

for any integer i, 1 ≤ i ≤ m, then each manifold M ,∀isometrically embedded into R (n i ) and for∀1∈− 1(n i),∀

can be isometrically embedded into R (n i )∩R (n j ) if M 1 ∩MNotice that in this case, serval manifolds in M (n1, n2, · · ·, nm

embedded into one Euclidean space R (n i ) for any integeapplying Lemma 5 .2 we know that M (n

1, n

2,

· · ·, n

m) can be

into a combinatorial Euclidean space R (k1, · · ·, kl).

Similar to the proof of Corollary 6 .3.1, we can get a for isometry embedding of combinatorial Riemannian manifoEuclidean spaces.

Corollary 6.3.2 A combinatorial Riemannian manifold Misometry embedded into a combinatorial Euclidean space R (

(i) l ≥m;

(ii ) there exists m different integers ki1 , ki2 , · · ·, kim ∈



(ii ) there exists m different integers ki1 , ki2 , · · ·, kim ∈

ki j ≥n2 j + 10 n j + 3and

dim (R ki j R ki r ) ≥dim 2(M n j M n r ) + 10dim (M

for any integer i,j, 1 ≤ i, j ≤m with M n j ∩M n r = ∅.




where detM is the determinant of M . It can be shown that Ggroup. In fact, since the function det : M n × n →R is contopen in R n 2 , and hence an open subset of R n 2 .

We show the mappings φ : GL(n, R

×GL(n, R ))

→GL(

GL(n, R ) determined by φ(a, b) = ab and ψ(a) = a− 1 aa, b∈GL(n, R ). Let a = ( a ij )n × n and b = ( bij )n × n ∈M (nknow that

ab = (( ab)ij ) = (n

k=1

a ik bkj ).

Whence, φ(a, b) = ab is continuous. Similarly, let ψ(a) = ( ψthat

ψij =a∗ij

deta

is continuous, where a∗

ij is the cofactor of a ij in the determGL(n, R ) is a topological group.

Now for integers n1, n2, · · ·, nm ≥1, let E G (GLn 1 , · · ·, Gconsisting of GL(n1, R ), GL(n2, R ), · · ·, GL(nm , R ) undestructure G. Then it is itself a combinatorial space. Whence



is a topological multi-group.A topological space S is homogenous if for∀a, b∈S , th

mapping f : S →S such that f (b) = a. We have the next re

Theorem 6.4.1 If a topological multi-group (S G ; O ) is arcsociative, then it is homogenous.


is well-dened and

a ◦0 c1 ◦1 c− 11 ◦2 c2 ◦3 c3 ◦4 · · · ◦s− 1 c− 1

s ◦s b− 1 ◦s b =

Let L = a ◦0 c1 ◦1 c− 11 ◦2 c2 ◦3 c3 ◦4 · · · ◦s− 1 c

− 1s ◦s b

− 1

◦s . Thenby the denition of topological multi-group. We nally get a conL : S G →S G such that L(b) = Lb = a. Whence, (S G ; O ) is hom

Corollary 6.4.1 A topological group is homogenous if it is arcwi

A multi-subsystem ( L H ;

O) of (S G ; O ) is called a topologi

if it itself is a topological multi-group. Denoted by L H ≤ S G

topological multi-subgroups is shown in the following.

Theorem 6.4.2 A multi-subsystem (L H ;O1) is a topological (S G ; O ), where O1 ⊂ Oif and only if it is a multi-subgroup obra.

Proof The necessity is obvious. For the sufficiency, we only for any operation ◦ ∈ O1, a ◦b− 1 is continuous in L H . Notice (iii ) in the denition of topological multi-group can be replaced b

for any neighborhood NS (a b− 1) of a b− 1 in S G there alw



for any neighborhood N S G (a ◦b ) of a ◦b in S G , there alw

hoods N S G (a) and N S G (b− 1

) of a and b− 1

such that N S G (a) ◦N Sb− 1), where N S G (a) ◦N S G (b− 1) = {x ◦y|∀x∈N S G (a), y∈N S G (b

by the denition of mapping continuity. Whence, we only need any neighborhood N L H (x ◦y− 1) in L H , where x, y ∈

L H and ◦neighborhoods N L H (x) and N L H (y− 1) such that N L H (x)◦N L H (y−


Corollary 6.4.2 A subset of a topological group (Γ; ◦) is aand only if it is a subgroup of (Γ; ◦) in algebra.

For two topological multi-groups ( S G1 ; O 1) and (S G

(S

G1 ;O

1) →(S

G2 ;O

2) is a homomorphism if it satises the (1) ω is a homomorphism from multi-groups ( S G1 ; O 1

for∀a, b∈S G1 and ◦∈O1, ω(a ◦b) = ω(a)ω(◦)ω(b);

(2) ω is a continuous mapping from topological spaces

∀x∈S G1 and a neighborhood U of ω(x), ω− 1(U ) is a neigh

Furthermore, if ω : (S G1 ; O 1) →(S G2 ; O 2) is an isoma homeomorphism in topology, then it is called an isomorautomorphism if (S G1 ; O 1) = ( S G2 ; O 2) between topological and ( S G2 ; O 2).

Let (S G ; O ) be an associatively topological multi-subgro

its topological multi-subgroups with S G =

m

i=1 H i , L H =According to Theorem 2 .3.1 in Chapter 2, for any integera quotient group H i / G i , i.e., a multi-subgroup ( S G / L H ;

algebraic multi-groups.Notice that for a topological space S with an equivalen



p g p q

jection π : S →S/ ∼= {[x]|∀y∈[x], y∼x}, we can introduby dening its opened sets to be subsets V in S/ ∼such thS . Such topological space S/ ∼is called a quotient space . N(S G ; O ) by a∼b for a, b∈

S G providing b = h ◦a for an eoperation ◦∈O. It is easily to know that such relation is an

l i d d i S / L


Kerω = {a∈S G1 | ω(a) = 1 ◦ ∈ I (O 2) }.

Proof According to Theorem 2 .3.2 or Corollary 2.3.1, we kn

representation pairs ( R1, P 1) and (R2, P 2) such that

(S G1 ; O 1)(Kerω; O 1) |(R 1 ,P 1 )

σ

∼=(S G2 ; O 2)

( I (O2); O2) |(R 2 ,P 2 )

in algebra, where σ(a ◦Kerω) = σ(a) ◦− 1 I (O 2) in the proof of Tonly need to prove that σ and σ− 1 are continuous.

On the First, for x = σ(a) ◦− 1 I (O 2)∈(S G 2 ;O 2 )

( I (O2 );O2 ) |(R 2 ,P 2 ) let U b

of σ− 1(x) in the space (S G 1 ;O 1 )(Ker ω;O 1 ) |(R 1 ,P 1 ) , where U is a union of a ◦

opened set U and ◦ ∈P 1. Since ω is opened, there is a neighborthat ω(U )⊃V , which enables us to nd that σ− 1(

V )⊂U . In fac

there exists y∈U such that ω(y) =

y. Whence, σ− 1(

y) = y ◦Ker

σ− 1 is continuous.On the other hand, let V be a neighborhood of σ(x) ◦− 1 I(S G 2 ;O 2 )

( I (O2 );O2 ) |(R 2 ,P 2 ) for x ◦ Kerω. By the continuity of ω, we kno

neighborhood U of x such that ω(U ) ⊂V . Denoted by U thez ◦Kerω for z ∈U. Then σ(U) ⊂V because of ω(U) ⊂V.



z ◦Kerω for z ∈U . Then σ(

U ) ⊂V because of ω(U ) ⊂V .

continuous. Combining the continuity of σ and its inverse σ− 1

, walso a homeomorphism from topological spaces (S G 1 ;O 1 )

(Ker ω;O 1 ) |(R 1 ,P 1 ) to

Corollary 6.4.3 Let ω : (S G ; O ) →(A ;◦) be a onto homomorphiical multi-group (S G ; O ) to a topological group (A ;◦). Then there pairs (R, P ), P ⊂

O such that


(i) (H i ; + i , ·i) is a ring for each integer i, 1 ≤ i ≤m, multi-ring;

(ii ) A is a combinatorial topological space S G ;

(iii ) the mappings ( a, b) →a ·i b− 1

, (a, b) →a + i (−∀a, b∈

H i , 1 ≤ i ≤m.

Denoted by ( S G ; O 1 →O 2) a topological multi-ring. A(S G ; O 1 →O 2) is called a topological divisible multi-ring or mcondition ( i) is replaced by (H i ; + i , ·i) is a divisible ring o

1 ≤ i ≤m. Particularly, if m = 1, then a topological multi-rior multi-eld is nothing but a topological ring, divisible ring oof topological elds are presented in the following.

Example 6.4.3 A 1-dimensional Euclidean space R is a topitself a eld under operations additive + and multiplication

Example 6.4.4 A 2-dimensional Euclidean space R 2 is isomeld since for∀(x, y)∈R 2, it can be endowed with a unique cwhere i2 = −1. It is well-known that all complex numbers fo

Example 6.4.5 A 4-dimensional Euclidean space R 4 is isical eld since for each point (x y z w ) R 4 it can be e



ical eld since for each point (x,y,z,w )

∈

R , it can be e

quaternion number x + iy + jz + kw, where

ij = − ji = k, jk = −kj = i, ki = −ik =

and

i2 = j 2 = k2 = 1.


Proof The proof on this classication theorem is needed a cathe topological structure and nished by Pontrjagin in 1934. A this theorem can be found in references [Pon1] or [Pon2].

Applying Theorem 6 .4.4 enables one to determine these topolo

Theorem 6.4.5 For any connected graph G, a locally compacted eld (S G ; O 1 →O 2) is isomorphic to one of the following:

(i) Euclidean space R , R 2 or R 4 endowed respectively withor quaternion number for each point if |G| = 1 ;

(ii ) combinatorial Euclidean space E G (2, · · ·, 2, 4, · · ·, 4) withi.e., the dimensional number lij = 1 , 2 or 3 of an edge (R i , R j

i = j = 4 , otherwise lij = 1 if |G| ≥2.

Proof By the denition of topological multi-eld ( S G ; O 1 →i, 1 ≤ i ≤ m, (H i ; + i , ·i) is itself a locally compacted topologic

(S G ; O 1 →O 2) is a topologically combinatorial multi-eld concompacted topological elds. According to Theorem 6 .4.4, we kn

(H i ; + i , ·i)∼= R , R 2, or R 4

for each integer i 1 i m Let the coordinate system of R R



for each integer i, 1

≤i

≤m. Let the coordinate system of R , R

and (z1, z2, z3, z4). If |G| = 1, then it is just the classifying in Thelet |G| ≥2. For ∀(R i , R j )∈E (G), we know that R i \ R j = ∅anthe denition of combinatorial space. Whence, i, j = 2 or 4. If ilij = 1 because of 1 ≤ lij < 2, which means lij ≥2 only if i = j = the proof.


Notice that if m = 1, then a Lie multi-group L G is ngroup in classical differential geometry. For example, the toshown in Examples 6.4.1 and 6.4.2 are Lie multi-groups sincemapping ( a, b)

→a

◦b− 1 is C ∞ -differentiable for a, b

∈

A

of a ◦b− 1. Furthermore, we give an important example follow

Example 6.4.6 An n-dimensional special linear group

SL (n, R ) = {M ∈GL(n, R ) | detM =

is a Lie group. In fact, let det M : R n 2

→R be the determin

to show that for M ∈det − 1(1), d(det M ) = 0. If so, thenfunction theorem, i.e., Theorem 3 .2.6, SL (n, R ) is a smooth

Let M = ( a ij )n × n . Then

det M =π∈S n

signπ a1π (1) · · ·anπ (n ) .

whence, we get that

d(det M ) =n

j =1 π∈S n

signπ a1π (1) · · ·a j − 1π ( j − 1) a j +1 π ( j +1)



Notice that the coefficient in da ij of the (i, j ) entry in tnant of the cofactor of a ij in M . Therefore, they can not vof det− 1(1). Now since {da ij }is linearly independent, there So applying the implicit function theorem, we know that Ssubmanifold of GL(n, R ). Now let M G be a combinatorial


Denition 6.4.2 A vector eld X on a Lie multi-group L G is invariant for ◦∈O (L G ) if

dLgX (

◦, x) = X (Lg(

◦, x))

holds for ∀g, x ∈H ◦ and globally left-invariant if it is locally

∀◦∈O (L G) and ∀g∈A (L G ).

Theorem 6.4.6 A vector eld X on a Lie multi-group L G is lo for ◦∈O (L G ) (or globally left-invariant) if and only if

dLgX (◦, 1◦ ) = X (g)

holds for ∀g∈H ◦ (or hold for ∀g∈

A (L G ) and ∀◦∈O (L G)).

Proof In fact, let ◦ ∈O (L G ) and g∈H ◦ (or g∈

A (L G

left-invariant for ◦∈O

(L

G ), then we know thatdLgX (◦, 1◦ ) = X (Lg(◦, 1◦ )) = X (g ◦1◦ ) = X (g

by denition. Conversely, if

dLgX (◦, 1◦ ) = X (g)



holds for∀g∈H ◦ and ◦∈O (L G ), let x∈

H ◦ . We get hat

X (Lg(◦, x)) = X (g ◦x) = dLg◦ xX (◦, 1◦ )

= dLg ◦Lx (X (◦, 1◦ )) = dLg(dLx (X (◦


Recall that a Lie algebra over a real eld R is a pair (vector space and [ , ] : F ×F →F with (X, Y ) →[X, Y ] that

[a1X 1 + a2Y 2, Y ] = a1[X 1, Y ] + a2[X 2, Y

[X, a 1Y 1 + a2Y 2] = a1[X 1, Y 1] + a2[X 2, Y

for∀a1, a2∈R and X,Y,X 1, X 2, Y 1, Y 2∈F . By Theorem 5

[X, Y ] = 0,

[[X, Y ], Z ] + [[Y, Z ], X ] + [[Z, X ], Y ] =

for X,Y,Z ∈X (L G ). Notice that all vector elds in X (L

over R , where, for X, Y ∈X (L G ), p∈

L G , f ∈X p and λ, μ

and [X, Y ]

∈

X (L G ) are dened by (X + Y )f = Xf + Y[X, Y ]v = X (Y f ) −Y (Xf ).

Now for a ◦∈O (L G ), dene

Y (◦, L G ) = {X ∈X (L G ) | dLgu(◦, x) = X (Lg(◦,

and



Y (L G ) = {X ∈X (L G )|dLgX (◦, x) = X (Lg(◦, x)) ,∀◦∈O

i.e., the sets of all locally left-invariant vector elds for an opall globally left-invariant elds. We can easily check that Y (In fact,


Therefore, Y (◦, L G ) is a Lie algebra. By denition, we know tha

Y (L G ) =◦∈O

Y (◦, L G ).

Whence, Y (L G ) is also a Lie algebra by denition.

Theorem 6.4.7 Let L G be a Lie multi-group. Then the mapping

Φ :◦∈O

Y (◦, L G ) →◦∈O

T 1 ◦ (L G )

determined by Φ(X ) = X (1◦ ) if dLgX (◦, x) = X (Lg(◦, x)) foisomorphism of

◦∈O Y (◦, L G ) with direct sum of T 1 ◦ (L G ) to L G

◦∈O (L G ).

Proof For an operation ◦∈O (L G ), we show that Φ|H ◦ : Y (is an isomorphism. In fact, Φ

|H ◦ is linear by denition. If Φ

|H ◦ (X

for ∀g∈H ◦ , we get that X (g) = dLg(X (◦, 1◦ )) = dLg(Y (◦, 1◦ )

X = Y . We know Φ|H ◦ is injective.Let W ∈T 1 ◦ (H ◦ ). We can dene a vector eld X on

Lg(◦, W ) = X (g) for every g ∈H ◦ . Thus, X (1◦ ) = L1 ◦ W =

eld is left invariant. In fact, for g1, g2∈H ◦ , we have



∈

X (Lg1 (g2)) = X (g1g2) = dLg1 g2 (W ) = dLg1 ◦dLg2 (W ) = d

Therefore, for W ∈T 1 ◦ (H ◦ ), there exists a vector eld X ∈Y

Φ|H ◦ (X ) = W , i.e., Φ|H ◦ is surjective. Whence, Φ |H ◦ : Y (◦, L G )


determined by Φ(X ) = X (1G ) if dLgX (◦, x) = X (Lg(◦, x))morphism of Y (◦, G ) with T 1 G (G ) to G at identity 1G .

For nding local form of a vector eld X ∈X (L G ) o

at a point p

∈

L G , we have known that

X = [a ij ( p)]s ( p)× n s ( p ) , [∂

∂x]s ( p)× n s ( p ) =

s ( p)

i=1

n s (

j =

by Theorem 5.1.3, where xil = x jl for 1 ≤ i, j ≤ s( p), 1 ≤have the following result.

Theorem 6.4.8 Let L G be a Lie multi-group. If a vectolocally left-invariant for an operation ◦∈O (L G ), then,

X p =s( p)

i=1

n s ( p )

j =1

a ij ( p)∂

∂x ij

with

a ij (Lg(◦, p)) = j

a ij ( p)∂ Lg(◦, y)ij

∂y ij |y=

for g, p ∈L G . Furthermore, X is globally left-invariant



invariant for ∀◦∈O (L G ).Proof According to Theorem 5 .1.3, we know that

X (g ◦ p)f (y) = j

a ij (g ◦ p)∂f (y)∂y ij |y= g


∂ (f (Lg)(◦, y))∂y ij |y= p =

s

∂f (g ◦y)∂ (g ◦y)is

∂ (g ◦y)is

∂y ij |y= p

=s

∂f (y)∂y is |y= g◦ p

∂ (g

◦y)is

∂y ij |y

By assumption, X is locally left-invariant for ◦. We know th(dLgX ) p(◦, f (y)), namely,

j

a ij (g ◦ p) ∂f (y)∂y ij |y= g◦ p =

i

(s

a is ( p) ∂ (g ◦y)is

∂y is |y= p) ∂f∂

Whence, we nally get that

a ij (Lg(

◦, p)) =

j

a ij ( p)∂ Lg(◦, y)ij

∂yij

|y= p

Example 6.4.7 Let R(n1, · · ·, nm ) be a combinatorial Euclideanof R n 1 , · · ·, R n m . It is a Lie multi-group by verifying each operatin Example 6.4.1 is C ∞ -differentiable. For this combinatorial spac



invariant Lg for + i is

Lg(+ i , p) = g + i p.

Whence, a locally left-invariant vector eld X must has a form

m n


6.4.3 Homomorphism on Lie Multi-Group. Letmulti-groups. A topological homomorphism ω : L G1 →morphism on Lie multi-group if ω is C ∞ differentiable. E (GL(n1, R ), GL(n2, R ),

· · ·, GL(nm , R )), then a homomorp

is called a multi-representation of L G1 .Now let Y i be one Lie algebra of L G i for i=1 or 2. A

is a Lie algebra homomorphism if it is linear with

[X, Y ] = [ (X ), (Y )] for ∀X, Y ∈

Particularly, if Y 2 = Y (GL(n, R )) in case, then a Lie algebrcalled a representation of the Lie algebra Y 1. Furthermore,isomorphism, then it is said that Y 1 and Y 2 are isomorphic

Notice that if ω : L G1 →L G2 is a homomorphism on Lieω maps an identity 1 ◦ of L G1 to an identity 1 ω(◦ ) of L G2 for anWhence, the differential dω of ω at 1 ◦

∈

L G1 is a linear trainto T 1ω ( ◦ )L G2 . By Theorem 6.4.7, dω naturally induces a lin

dω : Y 1 →Y 2

between Lie algebras on them. We know the following result

Theorem 6.4.9 The induced linear transformation dω : Y 1



1

homomorphism.

Proof For ∀X, Y ∈X (L G1 ) and f ∈

X p, we know tha

(dω[X Y ]f )ω = [X Y ](fω ) = X (Y(fω))


diffeomorphism, i.e., if a, b∈U 1◦ , then a◦b∈U 1◦ if and only if ω(a

with ω(a ◦b) = ω(a)ω(◦)ω(b), denoted by L LG1

ω

∼= L LG2 . Similarly, i

momorphism : Y 1 →Y 2 is an isomorphism, then it is said thatto Y 2, denoted by Y 1 ∼= Y 2. For Lie groups, we know the folloby Sophus Lie himself.

Theorem 6.4.10(Lie) Let Y i be a Lie algebra of a Lie group G i

G L1ω

∼= G L2 if and only if Y 1dω

∼= Y 2.

This theorem is usually called the fundamental theorem of L

us knowing that a Lie algebra of a Lie group is a complete invastructure of this group. For its a proof, the reader is refereed to re[Pon1] or [Var1] for examples. Then what is its revised form of theorem on Lie multi-groups? We know its an extended form onfollowing.

Theorem 6.4.11 Let Y (◦, L G i ) be a Lie algebra of a Lie mult

◦ ∈O (L G i ), i = 1 , 2. Then L LG1

ω

∼= L LG2 if and only if Y (◦, L G1

for ∀◦∈O (L G1 ).

Proof By denition, if L LG1

ω

∼= L LG2 , then for ◦∈O (L G1 ), th



dω : Y (◦, L G1 ) →Y (ω(◦), L G2 )

is an isomorphism by Theorem 6 .4.9. Whence, Y (◦, L G1 )dω

∼= Y (ωO (L G1 ).

Conversely if Y ( L G )dω

∼= Y (ω( ) L G ) for∀ ∈O (L G ) b


ad◦ (a)X i = di◦a X i =

s

j =1a ji (a) ◦X j .

By Theorem 6.4.9, the differential of the mapping ad ◦ (a) : L

an adjoint representation of Y (◦, L G), denoted by Ad◦ : Y (◦Then we know that

Ad◦ (X ) ◦Y = X ◦Y −Y ◦X = [X, Y

in the references, for example [AbM1] or [Wes1].

6.4.5 Lie Multi-Subgroup. A Lie multi-group L H is calof L G if

(i) L H is a smoothly combinatorial submanifold of L G

(ii ) L H is a multi-subgroup of L G in algebra.

Particularly, if L H is a Lie group, then we say it to be a L

well-known result is due to E.Cartan .

Theorem 6.4.12(Cartan) A closed subgroup of a Lie group

The proof of this theorem can be found in references, [Var1]. Based on this Cartan’s theorem, we know the followin



subgroups.Theorem 6.4.13 Let L G be a Lie multi-group with condihold, where A (L G ) =

m

i=1H i and O (L G ) =

m

i=1 {◦i}. Then a m

of L G is a Lie multi-group if

(i) (H ;O)|◦ i is a closed subgroup of (A ; O )|◦ i for any


where R i = R and + i = +. A homomorphism ϕ : R →L G oni.e., for an integer i, 1 ≤ i ≤m and ∀s, t ∈R , ϕ(s + i t) = ϕ(s)◦iϕ

parameter multi-group . Particularly, a homomorphism ϕ : R →L

parameter subgroup , as usual. For example, if we chosen a

◦ ∈O

the one-parameter multi-subgroup of L G is nothing but a one parof (H ◦ ;◦). In this special case, for ∀X, Y ∈

X (M ) we can deneLX Y of Y with respect to X introduced in Denition 5 .7.2 by

LX Y (x) = limt→0

1t

[ϕ∗t Y (ϕt(x)) −Y (x)]

for x∈M , where {ϕt}is the 1-parameter group generated by Xthat this denition is equivalent to Denition 5 .7.2, i.e., LX Y = X

Notice that ( R ; +) is commutative. For any integer i, 1 ≤ i ≤ϕ(t) ◦i ϕ(s) = ϕ(s + i t) = ϕ(s) ◦i ϕ(t), i.e., {ϕ(t), t∈R}is a commof (H ◦ i ;◦i). Furthermore, since ϕ(0)◦iϕ(t) = ϕ(0+ i t) = ϕ(t), mu

on the right, we get that ϕ(0) = 1 ◦ i . Also, by ϕ(t) ◦i ϕ(−i t) =ϕ(t −i t− 1

+ i ) = ϕ(0+ i ) = 1 ◦ i , we have that ϕ− 1◦ i (t) = ϕ(−i t).

Notice that we can not conclude that 1 ◦ 1 = 1 ◦ 2 = · · ·= ϕ(0+ 2 ) = · · ·= ϕ(0+ m ) in the real eld R . In fact, we should havϕ(0+ 1 ) = ϕ(0+ 2 ) = · · · = ϕ(0+ m ) in the multi-space R by de



should be 1◦ 1 = 1 ◦ 2 = · · ·= 1 ◦ m .The existence of one-parameter multi-subgroups and one-para

of Lie multi-groups is obvious because of the existent one-paramLie groups. In such case, each one-parameter subgroup ϕ : Rwith a unique left-invariant vector eld X ∈Y (G ) on a Lie grou


subgraph of G, and G[ϕ(R )] = G if and only if for any inteH i ∩H j = ∅implies that there exist integers s, t such that ϕ(R ; + j ) with ϕ(t) ◦i ϕ(s) = ϕ(t) ◦ j ϕ(s) holds;

(ii ) if ϕ : R

→L G is a one-parameter subgroup, i.e., R

integer i0, 1 ≤ i0 ≤m such that ϕ(R )≺(H i0 ;◦i0 ).

Proof By denition, each ϕ(R , + i) is a commutative sany integer 1 ≤ i ≤ m. Consequently, ϕ(R ) is a commutaL G . Whence, G[ϕ(R )] is a subgraph of G by Theorem 2 .1.1

Now if G[ϕ(R )] = G, then for integers i, j , H i

∩H j =

∅ϕ(R ; + j ) = ∅. Let ϕ(s),ϕ(t)∈ϕ(R ; + i) ∩ϕ(R ; + j ). Thenϕ(t) = ϕ(s + i t) = ϕ(s + t) = ϕ(s + j t) = ϕ(s) ◦ j ϕ(t). That

The conclusion ( ii ) is obvious by denition. In fact, ◦i0

Let ϕ : R →L G be a one-parameter multi-subgroup Theorem 6 .4.15, we can introduce an exponential mapping e

exp :◦∈O (L G )

Y (◦, L G) ×O (L G ) →L

determined by



exp(X, ◦) = ϕX (1◦ ).

We have the following result on the exponential mappin

Theorem 6.4.16 Let ϕ : R →L G be a one-parameter muO (L ) ith ϕ(+)


Furthermore, we know that dϕtX ( dds ) = tX . Therefore, ϕtX = ϕX

let s = 1, we nally get that

exp(tX,

◦) = ϕtX (1◦ ) = ϕX (t),

which is the equality ( i).For ( ii ), by the denition of one-parameter subgroup, we kno

(exp( t1X, ◦)) ◦ (exp( t2X, ◦)) = ϕX (t1) ◦ϕX (t2) = ϕX

= exp(( t1 + t2)X, ◦)

and

exp(t− 1+ X, ◦) = ϕX (t− 1

+ ) = (ϕX (t))− 1+ = exp − 1(tX,

For an n-dimensional R -vector space V , L G is just a Lie grthis case, we can show that

exp(tX, ◦) = etX =∞

i=0

(tX )i

n!,



where X i =i

X ◦ · · · ◦X for X ∈Y (GL(n, R )). To see it make righthand side converges, we show it converges uniformly for X inof GL(n, R ). In fact, for a given bounded region Λ, by denition N > 0 such that for any matrix A = ( xij (A)) in this region


Example 6.4.8 Let the matrix X to be

X =⎡

⎢⎢⎣

0 −1 01 0 0

0 0 0

⎤

⎥⎥⎦

.

A direct calculation shows that

etX = I 3× 3 + tX +t2X 2

2!+

t3X 3

3!+ · · ·

= I 3× 3 + t⎡⎢⎢⎣

0 −1 01 0 00 0 0⎤⎥⎥⎦

+t2

2!⎡⎢⎢⎣−1 0 00 −1 00 0 0⎤⎥⎥⎦

+t3

3

= ⎡

⎢⎢⎣

(1 − t2

2! + · · ·) −(t − t3

3! + · · ·) 0(t − t3

3! + · · ·) (1 − t2

2! −· · ·) 0

0 0 1

⎤

⎥⎥⎦= ⎡⎢⎢⎣

cos t −sin t 0sin t cos t 0

0 0 1

⎤⎥⎥⎦

.

For a Lie multi-group homomorphism ω : L G1 →L

b d d L f ll



between ω,dω and exp on a ◦∈L G1 following.

Theorem 6.4.17 Let ω : L G1 →L G2 be a Lie multi-grouω(◦) = •∈O (L G2 ) for ◦∈O (L G1 ). Then the following dia

L G1L G2

¹

ω


exp(tdω(X ), ◦) is the unique one-parameter subgroup of L G2 witdω(X )(1 ◦ ). Consequently, ω(exp( tX, ◦)) = exp( tdω(X ), ◦) for ∀ω(exp(X, ◦)) = exp( dω(X ), ◦).

6.4.7 Action of Lie Multi-Group. We have discussed the actimulti-groups on nite multi-sets in Section 2 .5. The same idea cato innite multi-sets.

Let M be a smoothly combinatorial manifold consisting of ma

· · ·, M m and L G a Lie multi-group with ( A (L G ); O (L G)), where

and O (L G) =m

i=1 {◦i}. The Lie multi-group L G is called an actiondifferentiable mapping φ : L G ×M ×O (L G ) →M determined byfor g∈

H i , x∈M i , 1 ≤ i ≤m such that

(i) for∀x, y∈M i and g∈H i , g ◦i x, g ◦i y∈g ◦i M i a man

(ii ) (g1 ◦i

g2)

◦ix = g

1 ◦i(g

2 ◦ix) for g

1, g

2∈

H i;

(iii ) 1◦ i ◦i x = x.

In this case, the mapping x →g ◦x for ◦∈O (L G ) is a diffeon M . By denition, we know that g− 1

◦ ◦(g◦x) = g◦(g− 1◦ ◦x) = 1

x →g◦x is a diffeomorphism on M . We say L G is a faithful acting

f ∀∈H i li h 1 I i i f h



for∀x∈H ◦ implies that g = 1 ◦ . It is an easy exercise for the rea

no xed elements in the intersection of manifolds in M for a faiton M . We say L G is a freely acting on M if g ◦x = x only hold

Dene (L G )◦x0 = {g ∈

L G |g ◦ x0 = x0}. Then ( L G )◦x0 for

(L G ). In fact, if g ◦ x0 = x0, we nd that g− 1◦ ◦ (g ◦ x0) = g−

◦


Corollary 6.4.6 Let G be a Lie group, x∈G . Then G gx =

Analogous to the nite case, we say that L G acts tr

∀x, y ∈M , there exist elements g∈L G and ◦ ∈O (L G ) s

smoothly combinatorial manifold M is called a homogeneousif there is a Lie multi-group L G acting transitively on M . Ifhomogeneous combinatorial manifold is also called a homogwe have a structural result on homogeneous combinatorial m

Theorem 6.4.19 Let M be a smoothly combinatorial ma

multi-group L G acts. Then M is homogeneous if and only ihomogeneous.

Proof If M is homogeneous, by denition we know thon M , i.e., for ∀x, y ∈M , there exist g∈

L G and an intethat y = g ◦i x. Particularly, let x, y∈M i . Then we know

L G|H i = ( H i , ◦i) is transitive on M i , i.e., M i is a homogeneConversely, if each manifold M in M is homogeneous, i

transitively M , let x, y ∈M . If x and y are in one manifthere exists g∈

G M i with g : x →g ◦i x differentiable such x∈M i but y∈M j with i = j , 1 ≤ i, j ≤m, remember tha

th i th



there is a pathP (M i , M j ) = M k0 M k1 M k2 · · ·M kl M kl +

connecting M i and M j in GL [M ], where M k0 = M i , M kl +

M ki ∩M k i +1 , 0 ≤ i ≤ l. By assumption, there are elementN l G d h G h h


Theorem 6.4.20 Let M be a differentiable combinatorial manimanifolds M ◦ i , 1 ≤ i ≤ m, G ◦ i a Lie group acting differentiablon M ◦ i . Chosen xi ∈M ◦ i , a projection πi : G ◦ i →G ◦ i / (G ◦ i )x , ς i : G ◦ i →M ◦ i determined by ς i(g) = g ◦i x for g∈

G ◦ i induces a d

ς :m

i=1G ◦ i / (G ◦ i )x →

m

i=1M ◦ i

with ς = ( ς 1, ς 2, · · ·, ς m ) and ς iπi = ς i . Furthermore, ς is a diffeom

ς : L G / (L G )Δ →M,

where Δ = {xi , 1 ≤ i ≤m}and xi∈M i \ (M \ M i ), 1 ≤ i ≤m.

Proof For a given integer i, 1 ≤ i ≤m, let g∈G ◦ i . Then fo

have that g ◦i g ∈g ◦i (G ◦ i )x and ς (g ◦i g ) = ς (g). See the followirelation among these mappings ς i , π i and ς i .

¹

G ◦ i M ◦ i

G ◦ i / (G ◦ i )x

ς i

πiς i

Thus the mapping πi(g) = g i (G◦ i )x ςi(g) determines a mapping



Thus the mapping πi(g) = g◦i (G )x →ς i(g) determines a mappingM ◦ i with ς iπi(g) = ς i(g). Notice that πi : G ◦ i →G ◦ i / (G ◦ i )x inducestopology on G ◦ i / (G ◦ i )x by

U ⊂G ◦ i / (G ◦ i )x is open if and only if π − 1

i (U ) is open

1




Whence, a = d and b = −c. Consequently, we know that ad −bwhich means that

a bc d

=cos θ −sin θsin θ cos θ

,

i.e., a rigid rotation on R 2. Therefore, SL (2, R )i = SO (2, R ), group consisting of all 2 ×2 real orthogonal matrices of determina

§6.5 PRINCIPAL FIBRE BUNDLES

6.5.1 Principal Fiber Bundle. Let P , M be a differentiamanifolds and L G a Lie multi-group ( A (L G ); O (L G )) with

P =

m

i=1 P i , M =

s

i=1 M i , A (L G ) =

m

i=1H ◦ i , O (L G ) =

A differentiable principal ber bundle over M with group L G

ferentiably combinatorial manifold P , an action of L G on P sconditions PFB1-PFB3:

PFB1 For any integer i 1 ≤ i ≤ m H ◦ i acts different



PFB1. For any integer i, 1 ≤ i ≤ m, acts differentright without xed point, i.e.,

(x, g)∈P i ×H ◦ i →x ◦i g∈P i and x ◦i g = x implies tha

Sec.6.5 Principal Fiber Bundles

PFB3. For any integer i, 1 ≤ i ≤ m, P ∈Π− 1(M ◦M ◦ i , i.e., any x∈M ◦ i has a neighborhood U x and a diffeomU x ×L G with

T |Π− 1i (U x ) = T xi : Π− 1

i (U x ) →U x ×H ◦ i ; x →T xi (x)

called a local trivialization (abbreviated to LT) such that

∀g∈H ◦ i , (x)∈

H ◦ i .

We denote such a principal bre bundle by P (M, L G ). If

P (M, H ), the common principal ber bundle on a manifoldtence of P (M, L G ) is obvious at least for m = 1.

For an integer i, 1 ≤ i ≤ m, let T ui : Π− 1i (U u ) →U u ×

U v ×H ◦ i be two LTs of a principal ber bundle P (M, L G ). from T ui to T vi is a mapping iguv : U u ∩U v →H ◦ i dened by

for∀x = Π i( p)∈U u ∩U v.Notice that iguv (x) is independent of the choice p∈Π−

i

u ( p◦i g) ◦i− 1v ( p◦i g) = u ( p) ◦i g ◦i ( v( p) ◦i g)− 1

= u ( p) ◦i g ◦i g− 1◦ i ◦i

− 1v ( p) =

Wh th liti f ll i b i



Whence, these equalities following are obvious.

(i) iguu (z) = 1 ◦ i for∀z∈U u ;

(ii ) igvu (z) = ig− 1uv (z) for∀z∈U u ∩U v;

(iii ) i guv (z) ◦iigvw (z) ◦i

igwu (z) = 1 ◦ i for∀z∈Uu ∩Uv


Theorem 6.5.1 There is a natural correspondence between local trivializations.

Proof If Λ : U →P is a local section, then we dene T : Πfor integers 1

≤i

≤m by T xi (Λ(x)

◦i g) = ( x, g) for x

∈

U x

⊂

M i

Conversely, if T : Π− 1(U ) →U ×L G is a local trivializatisection Λ : U →P by Λ(x) = ( T ui )− 1(x, 1◦ i ) for x∈U x⊂M i .

6.5.2 Combinatorial Principal Fiber Bundle. A general waprincipal ber bundles P (M, L G ) over a differently combinatori

by a combinatorial technique, i.e., the voltage assignment α : Gnite group G . In Section 4.5.4, we have introduced combinato(M ∗, M,p, G) consisting of a covering combinatorial manifold M ∗

a combinatorial manifold M and a projection p : M ∗→M by the vα : GL [M ] →G . Consider the actions of Lie multi-groups on cofolds, we nd a natural construction way for principal ber bundlcombinatorial manifold M following.

Construction 6.5.1 For a family of principal ber bundles over mM l , such as those shown in Fig.6.5.1,

H ◦ 1H ◦ 2

H ◦ l



P M 1

M

P M 2

M

P M l

M

ΠM 1 ΠM lΠM 2




ΠM = Π |P M . We must have that Π M |P M ∩P M = Π |P M ∩P M =

Conversely, if for ∀(M , M ) ∈E (GL [M ]) and (P M , P M )E (GL [P ]), ΠM |P M ∩P M = Π M |P M ∩P M in P α (M, L G ), then Π M is a well-dened mapping. Other conditions of a principal b

veried immediately by Construction 6 .5.1.

6.5.3 Automorphism of Principal Fiber Bundle. In thethis book, we always assume P α (M, L G ) satisfying conditions i.e., it is a principal ber bundle over M . An automorphism ofdiffeomorphism ω : P

→P such that ω( p

◦i g) = ω( p)

◦i g for g

∈ p∈P ∈π − 1 (M i )P , where 1 ≤ i ≤ l.

Particularly, if l = 1, an automorphism of P α (M, L G ) with an vα : GL [M ] →Z 0 degenerates to an automorphism of a principal a manifold. Certainly, all automorphisms of P α (M, L G ) forms a Aut P α (M, L G ).

An automorphism of a general principal ber bundle P (M,duced similarly. For example, if ωi : P M i →P M i is an automomanifold M i for 1 ≤ i ≤ l with ωi|P M i ∩P M j

= ω j |P M i ∩P M jfor 1 ≤ i,

Gluing Lemma, there is a differentiable mapping ω : P

→P suc

for 1 ≤ i ≤ l. Such ω is an automorphism of P (M, L G ) by d



→≤ ≤ p ( , ) yconcentrate our attention on the automorphism of P α (M, L G ) combinatorially characterized.

Theorem 6.5.3 Let P α (M, L G ) be a principal ber bundle. The


ω( p◦i g) = hωi( p◦i g) = h(ωi( p) ◦i g) = hωi

for p∈P ∈π − 1 (M i )P and g∈

H ◦ i . Whence, ω is an automorph

A principal ber bundle P (M, L G ) is called to be normaexists an ω∈Aut P (M, L G ) such that ω(u) = v. We get the conditions of normally principal ber bundles P α (M, L G ) f

Theorem 6.5.4 P α (M, L G ) is normal if and only if P

(H

◦ i ;◦i) = (H

;◦) for 1 ≤ i ≤ l and GL α

[M ] is transitivtomorphisms in Aut GL α [M ].

Proof If P α (M, L G ) is normal, then for ∀u, v ∈PAut P α (M, L G ) such that ω(u) = v. Particularly, let u,i, 1 ≤ i ≤ l or GL α [M ]. Consider the actions of Aut P α (

Aut P α

(M, L G )|GL α [M ], we know that P M i (M i , H ◦ i ) for 1 ≤normal, and particularly, GL α [M ] is a transitive graph by dphisms in Aut GL α [M ].

Now choose u ∈M i and v ∈M j \ M i , 1 ≤ i, j ≤ l. an automorphism ω∈Aut P α (M, L G ) such that ω(u) = vω(u)

◦i g = v

◦i g by denition. But this equality is well-de

(H ◦ ;◦j ) Applying the normality of P α (M L G ) we nd t



( ◦ j ;◦ j ). Applying the normality of P (M, G ), we nd tfor any integer 1 ≤ i ≤ l.

Conversely, if P M i (M i , H ◦ i ) is normal, (H ◦ i ;◦i) = ( H

GL α [M ] is transitive by diffeomorphic automorphisms in Aut


Corollary 6.5.1 Let GL [M ] be a transitive labeled graph by diffeophisms in Aut GL [M ], α : GL [M ] →G a locally f -invariant voltagP (M, H ) a normal principal ber bundle. Then the constructed Ping each P M i (M i , H ◦ i ), 1 ≤ i ≤ l by P (M, H ) in Construction 6

Proof By Theorem 4.5.6, a diffeomorphic automorphism of GL α [M ]. According to Theorems 6.5.3 and 6.5.4, we know thanormally principal ber bundle.

6.5.4 Gauge Transformation. An automorphism ω of P α (induces a diffeomorphism ω : M

→M determined by ω(Π( p)) = Π

tion of ω motivates us to raise the conception of gauge transformattheoretical physics. A gauge transformation of a principal ber bis such an automorphism ω : P →P with ω =identity transformΠ( p) = Π( ω( p)) for p∈P . Similarly, all gauge transformations aldenoted by GA(P ).

There are many gauge transformations on principal ber bundlthe identity transformations 1 P M i

induced by the right action ofGL α [M ], i.e., h((M i )g) = ( M i )g◦ i h for∀h∈G , 1 ≤ i ≤ l are all suc

Let P α (M, L G ) be a principal ber bundle and ( H ◦ i ;◦i) aF i to the left, i.e., for each g ∈

H ◦ i , there is a C ∞ -mapping i

F i such that iL1◦ i (u, ◦i) = u and iLg1 ◦ i g2 (u, ◦i) = i Lg1 ◦i iLg2 (ui



Particularly, let F i be a vector space R n i and iLg a linear mappincase, a homomorphism H ◦ i →GL(n i , R ) determined by g →Lrepresentation of H ◦ i . Two such representations g →Lg and gto be equivalent if there is a linear mapping T : GL(n i , R ) →GL


Theorem 6.5.5 Let P α (M, L G ) be a principal ber bundle wα : GL [M ] →G and C i(P M i , H ◦ i ) with an action g(g ) =itself, 1 ≤ i ≤ l. Then

GA(P )∼= R(G) (l

i=1C i(P M i , H ◦ i ))

where R(G) denotes all identity transformations 1P M i, 1 ≤ i

action of G on vertices in GL α [M ].

Proof For any ∈C i(P M i , H ◦ i ), dene ω : P M i →P M

for u∈P M i . Notice that ω(u ◦i g) = u ◦i g ◦i (u ◦i g) =u ◦i (u) ◦i g = ω(u) ◦i g. It follows that ω∈GA(P M i ).

Conversely, if ω∈GA(P M i ), dene : P M i →H ◦ i

u ◦i (u). Then u ◦i g ◦ (u ◦i g) = ω(u ◦i g) = ω(u) ◦i g =(u ◦i g) = g− 1

◦ i ◦i (u) ◦i g and it follows that ∈C i(P

if ω, ω ∈GA(P M i ) with ω(u) = u ◦i (u) and ω (u) = u ◦u ◦i (τ (u)τ (u)). We know that GA(P M i )∼= C i(P M i , H ◦ i ).Extend such isomorphisms ι i : GA(P M i ) →C i(P M i , H ◦ i )

Notice that all identity transformations 1 P M iinduced by th

vertices in GL α [M ] induce gauge transformations of P α (Mget that

l



GA(P )⊇R(G) (l

i=1

C i(P M i , H ◦ i ))

Besides, each gauge transformation ω of P α (M, L G )b d d f b


For any integer i, 1 ≤ i ≤ l, let Y (L G , ◦i) be a Lie algebra oan adjoint representation ad◦ i : H ◦ i →GL(Y (L G , ◦i)) given by gH ◦ i . Then the space C i(P M i , Y (L G , ◦i)) is called a gauge algebrIf C i(P M i , Y (L G , ◦i)) has be dened for all integers 1 ≤ i ≤ l, the

l

i=1

C i(P M i , Y (L G , ◦i))

is called a gauge multi-algebra of P α (M, Y (L G )), denoted by C (

Theorem 6.5.6 For an integer i, 1

≤i

≤l, if H i , H i

∈

Clet [H i , H i ] : P M i →H ◦ i be a mapping dened by [H i , H i ](u) =

∀u∈P M i . Then [H i , H i ]∈C i(P M i , Y (L G , ◦i)) , i.e., C i(P M i , Y (algebra structure. Consequently, C (P , L G ) has a Lie multi-algebr


[H i , H i ](u ◦i g) = [H i(u ◦i g), H i (u ◦i g)]

= [ad◦ i (g− 1◦ i )H i(u), ad ◦ i (g− 1

◦ i )H i (u)]

= ad◦ i (g− 1◦ i )[H (u), H (u)] = ad◦ i (g− 1

◦ i )[H

for ∀u ∈P M i . Whence, C i(P M i , Y (L G , ◦i)) inherits a Lie algebC(P L ) h Li lti l b t t



∀∈C (P , L G ) has a Lie multi-algebra structure.

6.5.5 Connection on Principal Fiber Bundle. A local connpal ber bundle P α (M, L G ) is a linear mapping i Γu : T x (M ) →Tu

1 i


(ii ) ΓR g ◦ u = dRg ◦Γu for ∀g∈L G and ∀◦ ∈O (L G)

right translation on P ;

(iii ) the mapping u →Γu is C ∞ .

Certainly, there exist closed relations between the local aon principal ber bundles. A local or global connection on aP α (M, L G ) are distinguished by or not by indexes i for 1tion. We consider the local connections rst, and then the glofollowing.

Let iH u = iΓu (T x (M )) and iV u = T u (iF x ) the space of

ber i F x , x ∈M i at u ∈P M i with Π i(u) = x. Notice T x ({x}) = {0}. For∀X ∈

iV u , there must be dΠi(X ) = 0. Thare called horizontal or vertical space of the connection iΓu

Theorem 6.5.7 For an integer i, 1 ≤ i ≤ l, a local coassignment iH : u

→iH u

⊂

T u (P ), of a subspace iH u of T u (

(i) T u (P ) = iH u⊕i V u , u∈

i F x ;

(ii ) (d iRg) iH u = iH u ◦ i g for ∀u∈iF x and ∀g∈

H ◦ i ;

(iii ) i H is a C ∞ -distribution on P .

Proof By the linearity of the mapping i Γu , u

∈

i F x forsubspace of the tangent space T u (P ). Since (dΠi)iΓu = ident



we know that dΠi is one-to-one. Whence, dΠi : iH u →T Π( u)

which alludes that i H u ∩i V u = {0}. In fact, if iH u ∩iV u =X = 0. Then dΠiX = 0 and dΠiX ∈T x (M ). Because dΠi hi k th t K dΠ {0} hi h t di t


So the property ( ii ) holds. Finally, the C ∞ -differentiable of i H C ∞ -differentiable of the mapping u →iΓu .

Conversely, if iH : u →iH u is a such C ∞ distribution on Plocal connection to be a linear mapping iΓu : T x (M ) →T u (P ) for

x∈M i by iΓu (T u (M )) = iH u , which is a connection on P α (M, L

Theorem 6 .5.7(i) gives a projection of T u (P ) onto the tangeniF x with x∈M i and Π i(u) = x by

iv : T u (P ) →T u (iF x ); X = X v + X h →ivX = X

Moreover, there is an isomorphism from Y (H ◦ i , ◦i) to T u (iF x ) bwhich enables us to know that a local connection on a principal balso in terms of a Y -valued 1-forms.

Theorem 6.5.8 Let P α (M, L G ) be a principal ber bundle. Th

i, 1 ≤ i ≤ l,(i) there exists an isomorphism ι i : Y (H ◦ i , ◦i) →T u (iF x )

Πi(u) = x;

(ii ) if ι i (X ) = X v∈Y (H ◦ i , ◦i), then ι i((d iRg)X ) = ad ◦ i (g− 1

Proof First, any left-invariant vector eld X

∈

X (H ◦ i ) giveld X ∈

X (P M i ) such that the mapping Y (H ◦ i , ◦i) →P M i deter



∈is a homomorphism, which is injective. If X u = 0 ∈P M i for so

X = 0 ◦ i ∈Y (H ◦ i , ◦i). Notice that u ◦i g = iRgu = u◦i exp(tX ), gsame ber as u by denition of the principal ber bundle and Co

h h (H ) (i )




iω((d iRg)X ) = iω([(d iRg)X ]v) = iω((d iRg)X v) = ad◦ i (

For showing that i ω depends differentiably on u, it suffice

any C ∞

-vector eld X ∈P ,iω(X ) is a differentiable Y (H ◦ i , ◦i)-v

fact, X is C ∞ implies that iv(X ) : u →(ivX )u and ih(X ) : u →(C ∞ and since ivX is differentiable at u, so is X v = iω(X ).

Conversely, given a differentiable Y (H ◦ i , ◦i)-valued 1-form i

ditions ( i)-(ii ) hold, dene the distribution

iH u = {X ∈T u (P ) | iω(X ) = 0 }.

Then the assignment u →iH u denes a local connection with itiω. In fact, for∀X ∈

iV u , iω(X ) = 0 implies X ∈i H u . Therefore

and T u (P ) = i H u + iV u . In fact, let iω(X ) = X v. But we know Let Z = X

−X v. We nd that iω(Z ) = iω(X )

−iω(X v) = 0.

which implies that T u (P ) = iH u⊕i V u . That is the condition ( i)

Now for any X ∈iH u , we have that iω((d iRg)X ) = ( ad

Whence, (d iRg)X is horizontal, i.e., ( d i Rg) i H u⊂iH u◦ i g.

Let X u◦ i g ∈iH u◦ i g with X u◦ i g = ( d iRg)X u for some X u ∈

that X u

∈

iH u . Notice that X u◦ i g = ( d iRg)X u is equivalent to X uWe get that



iω(X u ) = i ω((d iRg− 1 )X u◦ i g) = ( ad◦ i g− 1) iω(X u◦ i g)

which implies that X u is horizontal. Furthermore, since u →iω


(ii ) (dRg)H u = H u◦ g for ∀u∈F x , ∀g∈L G and ◦∈O

(iii ) H is a C ∞ -distribution on P .

Theorem 6.5.11 Let Γ be a global connection on P α (M, L

a Y (L G )-valued 1-form ω on P , i.e., the connection form salowing:

(i) ω(X ) is vertical, i.e., ω(X ) = ω(X v) = X v, whereω(X ) = 0 if and only if X ∈H u ;

(ii ) ω((dRg)X ) = ad ◦ g− 1ω(X ) for ∀g∈L G , ∀X ∈

X (

Certainly, all local connections on a principal ber buconnection on this principal ber bundle exist rst. But the coSo it is interesting to nd conditions under which a global know the following result on this question.

Theorem 6.5.12 Let iΓ be a local connections on P α (M, L

a global connection on P α (M, L G ) exists if and only if (H ◦ i

is a group and i Γ|M i ∩M j = j Γ|M i ∩M j for (M i , M j )∈E (GL [M

Proof If there exists a global connection Γ on a principal then Γ |M i , 1 ≤ i ≤ l are local connections on P α (M, L G ) witfor (M i , M j )∈E (GL [M ]), 1 ≤ i, j ≤ l.

Furthermore, by the condition ( ii ) in the denition of glou g is well dened for∀g∈

L∀ ∈

O (L ) i e g acts



u ◦g is well-dened for∀g∈L G , ∀◦ ∈O (L G ), i.e., g acts

(H ◦ i ;◦i) = ( H ;◦) if g∈H ◦ i , 1 ≤ i ≤ l, which means that L

Conversely, if L G is a group and iΓ|M i ∩M j = j Γ|M i ∩M j fo1 i j l we can dene a linear mapping Γ : T (M )


We have known there exists a connection on a common pridle P (M, H ) in classical differential geometry. For example, theor [Wes1]. Combining this fact with Theorems 6 .5.4 and 6.5.12consequence.

Corollary 6.5.3 There are always exist global connections on a n ber bundle P α (M, L G ).

6.5.6 Curvature Form on Principal Fiber Bundle. Letprincipal ber bundle associated with local connection form iωglobal connection form ω. A curvature form of a local or global a Y (H ◦ i , ◦i) or Y (L G )-valued 2-form

i Ω = (d i ω)h, or Ω = (dω)h,

where

(d iω)h(X, Y ) = d iω(hX,hY ), (dω)h(X, Y ) = dω(h

for X, Y ∈X (P M i ) or X, Y ∈

X (P ). Notice that a 1-form ωh(Xonly if ih(X 1) = 0 or ih(X 12) = 0. We have the following strucprincipal ber bundles.

Theorem 6.5.13(E.Cartan) Let iω, 1 ≤ i ≤ l and ω be local orforms on a principal ber bundle P α (M L ) Then



forms on a principal ber bundle P α (M, L G ). Then

(d i ω)(X, Y ) = −[ i ω(X ),i ω(Y )] + iΩ(X, Y )


The proof for the structural equation of global conneconsider three cases following.

Case 1. X , Y ∈iH u

In this case, X, Y are horizontal. Whence, iω(X ) = i ωwe know that ( d iω)(X, Y ) = i Ω(X, Y ) = −[ iω(X ),i ω(Y )] +

Case 2. X , Y ∈iV u

Applying the equation in Theorem 5 .2.5, we know that

(d iω)(X, Y ) = X iω(Y )

−Y iω(X )

−iω([

Notice that iω(X ) = i ω(X v) = X is a constant function. WY iω(X ) = 0. Hence,

(d iω)(X v, Y v) = − iω([X v, Y v]) = − i ω([X, Y ]v) =

[X

which means that the structural equation holds.

Case 3. X ∈iV u and Y ∈

iH u

Notice that i ω(Y ) = 0 and Y i ω(X ) = 0 with the samOne can shows that [ X, Y ]∈

iH u in this case. In fact, letwhere ϕt is the 1-parameter subgroup of H ◦ i generated by X

[X, Y ] = LX Y = limt→0

1t

(d iRϕt Y −Y )



implies that [ X, Y ] ∈iH u since Y and (d iRϕt )Y are h

6.5.10(ii ). Whence, iω([X, Y ]) = 0. Therefore, ( d iω)(X,tent with the right hand side of the structure equation.


(d iΩ)h = 0 , and (dΩ)h = 0 .

Proof We only check that ( d iΩ)h = 0 since the proof for (d

applying Theorem 6 .5.13, by denition, we now that

(d iΩ)h(X,Y,Z ) = dd i ωh(X,Y,Z ) +12

d[ iω, iω]h(X,Y

because of

dd iωh(X,Y,Z ) = 0 , and d[ iω, iω]h(X,Y,Z ) =

by applying Theorem 5 .2.4 and iω vanishes on horizontal vectors.

§6.6 REMARKS

6.6.1 Combinatorial Riemannian Submanifold. A combis a combination of manifolds underlying a connected graph Gto characterize its combinatorial submanifolds by properties of itsmanifolds. In fact, a special kind of combinatorial submanifoldsrial in-submanifolds are characterized by such way, for example, th

etc. in Section 4.2.2. Similarly, not like these Gauss ’s, Codazzi’ slae in Section 6.1, we can also describe combinatorial Riemannian



,such way by formulae on submanifolds of Riemannian manifolds a connected graph. This will enables us to nd new characters Riemannian submanifolds

Sec.6.6 Remarks

equations should be produced. Even through, these Gaussequations can be also seen as a kind of geometrical equationimportant in physics.

6.6.3 Embedding. By the Whitney’s result on embedding Euclidean space, any manifold is a submanifold of a Euclideageneralizes this result to combinatorial Riemannian submanifanswers a question in [Mao12]. Certainly, a combinatorial fold can be embedded into some combinatorial Euclidean spTheorem 6.3.7 with its corollary. Even through, there are m

on embedding a combinatorial Riemannian manifold or genemanifold into a combinatorial Riemannian manifold or a smmanifold. But the fundamental is to embed a smoothly cinto a combinatorial Euclidean space. For this objective, Theelementary such result.

6.6.4 Topological Multi-Group. In modern view pointa union of a topological space and a group, i.e., a Smarandmultiple 2. That is the motivation introducing topological mical multi-rings or topological multi-elds. The classicationtopological elds, i.e., Theorem 6 .4.4 is a wonderful result

mathematician Pontrjagin in 1930s. This result can be gemulti-spaces, i.e., Theorem 6 .4.5.



In topological groups, a topological subgroup of a topogroup of this topological group in algebra. The same is holdgroup Besides the most fancy thing on topological multi-gro


group operations differentiable. Certainly, it has similar properties aalso combinatorial behaviors. Elementary results on Lie groups angeneralized to Lie multi-groups in Section 6 .4. But there are stworks on Lie multi-groups should be done, for example, the repr

for Lie multi-groups, the classication of Lie algebras on Lie multi6.6.6 Principal Fiber Bundle. A classical principal ber buncombining of a manifold, its covering manifold associated with a Lit has been a fundamental conception in modern differential geomeThe principal ber bundle discussed in Section 6 .5 is an extende

sical, which is a Smarandachely principal ber bundle underlyingstructure G, i.e., a combinatorial principal ber bundle.

The voltage assignment technique α : GL →G is widely usedgraph theory for nd a regular covering of a graph G, particularlyof a graph in [GrT1]. Certainly, this kind of regular covering GLα

automorphisms, particularly, the right action R(G) on vertices of can be found in references, such as those of [GrT1], [MNS1], [Ma

Combining the voltage assignment technique α : GL →G wicipal ber bundles P M 1 (M 1, H ◦ 1 ), P M 2 (M 2, H ◦ 2 ), · · ·, P M l (M l , H

combinatorial principal ber bundles P α (M, L G ) in Constructio6.5 analogous to classical principal ber bundles. For example, themations are completely determined in Theorem 6 .5.5. The behavlikewise to classical principal ber bundles enables us to introdu



likewise to classical principal ber bundles enables us to introduor global Ehresmann connections, to determine those of local orforms, and to nd structure equations or Bianchi identity on su

CHAPTER 7.

Fields with Dynamics

All known matters are made of atoms and sub-atomic particlby four fundamental forces: gravity, electro-magnetism, stroand weak force, partially explained by the Relativity TheoField Theory . The former is characterized by actions in ex

later by actions in internal elds under the dynamics. Botcan be established by the Least Action Principle . For thistroduce variational principle, Lagrangian equations, Euler-Lagand Hamiltonian equations in Section 7 .1. In section 7.2, the and Einstein gravitational eld equations are presented, also s

nian eld to be that of a limitation of Einstein’s. Applying thmetric , spherical symmetric solutions of Einstein gravitationa



can be found in this section. This section also discussed tSchwarzschild geometry. For a preparation of the interaction, tromagnetism, such as those of electrostatic, magnetostatic a

348 Chap

§7.1 MECHANICAL FIELDS

7.1.1 Particle Dynamic. The phase of a physical particledetermined by a pair {x , v}of its position x and directed velogeometrical space P , such as those shown in Fig.7 .1.1.

¶

Ax

v

P

γ (t)

Fig. 7.1.1

If A is moving in a conservative eld Rn

with potential energy(x1(t), x2(t), · · ·, xn (t)) = γ (t) and

v = ( v1, v2, · · ·, vn ) =dxdt

= ( x1, x2, · · ·, xn )

at t. In other words, v is a tangent vector at v∈Rn, i.e., v∈T (

the force acting on A is



F = −∂U ∂x

= −(∂U ∂x 1

e1 +∂U ∂x 2

e2 + · · ·+∂U ∂x

en ).

Sec.7.1 Mechanical Fields

By denition, its momentum and moving energy are respectiv

p = mv = mx

and

T = 12

mv21 + 1

2mv2

2 + · · ·+ 12

mv2n = 1

2m

where v = |v |. Furthermore, if the particle A moves from ti

t2

t1

F ·dt = p |t2 −p |t1 = mv2 −mv1

by the momentum theorem in undergraduate physics.We deduce the Lagrange equations for the particle A. Fi

sides of (7−4) by dx = ( dx1, dx2, · · ·, dxn ) on, we nd that

−

n

i=1

∂U

∂x i

dx i =n

i=1

mxi dxi .

Let q = ( q1, q2, · · ·, qn ) be its generalized coordinates ofthat

xi = xi (q1, q2,

· · ·, qn ), i = 1 , 2,

· · ·, n.

Differentiating (7 −6), we get that



dx i =n

k=1

∂x i

∂qkdqk

350 Chap

Substitute (7 −8) and (7 −9) into (7 −5), we get that

n

k=1

(n

i=1

mxi∂x i

∂qk)dqk = −

n

i=1

∂U ∂qk

dqk .

Since dqk , k = 1 , 2,

· · ·, n are independent, there must be

n

i=1

mxi∂x i

∂qk= −

∂U ∂qk

, k = 1 , 2, · · ·, n.


n

i=1

mxi∂x i

∂qk=

ddt

(n

i=1

mxi∂x i

∂qk) −

n

i=1

mxiddt

∂x i

∂qk.

Substitute (7 −11) into (7 −10), we know that

ddt

(n

i=1

mxi∂x i

∂qk) −

n

i=1

mxiddt

∂x i

∂qk= −

n

i=1

∂U ∂qk

dqk

for k = 1 , 2, · · ·, n . For simplifying (7 −12), we need the differen∂x i /∂q k with respect to t following.

xi =dxi =

n ∂x i qk



xi dtk=1

∂qkqk ,


ddt

(n

i=1

mxi∂ xi

∂ qk) −

n

i=1

mxi∂ xi

∂qk= −

n

i=1

∂U∂qk

for k = 1 , 2,

· · ·, n . Because of the moving energy of A

T =12

mv2 =n

i=1

12

mx2i ,

partially differentiating it with respect to qk and qk , we nd

∂T ∂qk

=n

i=1

mxi∂ xi

∂qk, ∂T

∂ qk=

n

i=1

mxi∂ xi

∂ qk.

Comparing (7 −16) with (7 −17), we can rewrite (7 −16)

ddt

∂T ∂ qk −

∂T ∂qk = −

∂U ∂qk , k = 1 , 2, · · ·, n.

Since A is moving in a conservative eld, U (x) is independe∂U/∂ qk = 0 for k = 1 , 2, · · ·, n. By moving the right side to consequently get the Lagrange equations for the particle A f

ddt

∂ L∂ qk −

∂ L∂qk

= 0 , k = 1 , 2, · · ·, n,



where L= T −U is called the Lagrangian of A and

352 Chap

δF (K ) = F (K ) −F 0(K ).

For example, let K = [x0, x1], then we know that δf = f (xC [x0, x1], x∈[x0, x1] and δf (x0) = δf (x0) = 0, particularly, δx =

we furthermore know that

δdf dx

=df dx −

df 0dx

=d

dxδf,

i.e., [δ, ddx ] = 0. In mechanical elds, the following linear functiona

J [y(x)] = x

1

x0

F (x, y(x), y (x))dx

are fundamental, where y = dy/dx . So we concentrate our attenttionals and their variations. Assuming F ∈C [x0, x1] is 2-differentTaylor’s formula, then

Δ J = J [y(x) + δy]−J [y(x)]

= x1

x0

F (x, y(x) + δy,y (x) + δy )dx − x1

x0

F (x,

= x1

x0

(F (x, y(x) + δy,y (x) + δy )dx −F (x, y(x)

=x1

x0

(∂F ∂y

δy +∂F ∂y

δy )dx + o(D1[y(x) + δy,y(x)])



x0

The rst term in (7 −22) is called the rst order variation or just vadenoted by


Whence,

δF =∂F ∂y

δy +∂F ∂y

δy .

We can rewrite (7 −23) as follows.

δJ = δ x1

x0

F (x, y(x), y (x))dx = x1

x0

δF (x, y(x

Similarly, if the functional

J [y1, y2, · · ·, yn ] = x1

x0 F (x, y1, y2, · · ·, yn , y1, y2, · ·and F, y i , yi for 1 ≤ i ≤n are differentiable, then

δJ = x1

x0

δFdx = x1

x0

(n

i=1

∂F ∂yi

δyi +n

i=1

∂F ∂yi

δyi)

The following properties of variation are immediately go

(i) δ(F 1 + F 2) = δF 1 + δF 2;

(ii ) δ(F 1F 2) = F 1δF 2 + F 2δF 1, particularly, δ(F n ) = nF

(iii ) δ( F 1F 2 ) = F 2 δF 1 − F 1 δF 2

F 22;

(iv) δF (k) = ( δF )(k) , where f (k) = dkF/Dx k ;

(v) δ x1x F dx = x1

x δFdx.



( ) x0 x0

For example, let F = F (x, y(x), y (x)). Then

354 Chap

J [F (K )]−J [F 0(K )] ≥0 or ≤0 hold in a -neighborhood of F 0(K

called the maximal or minimal value of J [F (K )] in K . For suchwe have a simple criterion following.

Theorem 7.1.1 The functional J [y(x)] in (7-21) has maximal or

y(x) only if δJ = 0 .

Proof Let be a small parameter. We dene a function

Φ( ) = J [y(x) + δy] = x1

x0

F (x, y(x) + δy,y(x) +

Then J [y(x)] = Φ(0) and

Φ ( ) = x1

x0

(F (x, y(x) + δy,y(x) + δy)

∂yδy +

F (x, y(x) + δy,y∂y

Whence,

Φ (0) = x1

x0

( ∂F ∂y

δy + ∂F ∂y

δy )dx = δJ.

For a given y(x) and δy, Φ( ) is a function on the variable . BJ [y(x)] attains its maximal or minimal value at y(x), i.e., = 0. Bin calculus, there must be Φ (0) = 0. Therefore, δJ = 0.

7.1.3 Hamiltonian principle. A mechanical eld is dened to bΣ constraint on a physical law L , i.e., each particle in Σ is abidedlaw L , where Σ maybe discrete or continuous. Usually, L can



law , where Σ maybe discrete or continuous. Usually, can a system of functional equations in a properly chosen reference syalso describe a mechanical eld to be all solving particles of a sy


T =12

v, v , v∈T M ;

(iii ) A force eld given by a 1-form

ω =n

i=1ωidxi = ωidx i .

Denoted by T (M, ω) a mechanical eld. For determincal elds, there is a universal principle in physics, i.e., thepresented in the following.

Hamiltonian Principle Let T (M, ω) be a mechanical evariational S : T (M, ω) →R action on T (M, ω) whose truminimum value of S[(T (M, ω)], i.e., δS = 0 by Theorem 7.

In philosophy, the Hamiltonian principle reects a harthings developing in the universe, i.e., a minimum consumin

universe. In fact, all mechanical systems known by human principle. Applying this principle, we can establish classical mas those of Lagrange’s, Hamiltonian, the gravitational elds,

7.1.4 Lagrange Field. Let q(t) = ( q1(t), q2(t), · · ·, qn (t))dinate system for a mechanical eld T (M, ω). A Lagrange

with a differentiable Lagrangian L : T M →R , L= L(t, qNotice the least action is independent on evolving time of aL g g ld th i ti l ti i ll d t i d b



Lagrange eld, the variational action is usually determined by

356 Chap

Proof By (7 −25), we know that

δS = t2

t1

(n

i=1

∂ L∂qi

δqi +n

i=1

∂ L∂ qi

δqi)dt.

Notice that δqi = ddt δqi and

t2

t1

∂ L∂ qi

δqi )dt =∂ L∂ qi

δqi |t2t1 − t2

t1

ddt

∂ L∂ qi

δqidt

Because of δq(t1) = δq(t2) = 0, we get that

t2

t1

∂ L∂ qi

δqi)dt = − t2

t1

ddt

∂ L∂ qi

δqidt

for i = 1 , 2, · · ·, n . Substituting (7 −28) into (7 −27), we nd th

δS = t2

t1

n

i=1

(∂ L∂qi −

ddt

∂ L∂ qi

)δqidt.

Applying the Hamiltonian principle, there must be δS = 0i = 1 , 2, · · ·, n. But this can be only happens if each coefficient of that is,

∂ L∂qi −

ddt

∂ L∂qi

= 0 , i = 1 , 2, · · ·, n.



∂qi dt ∂ qi

These Lagrange equations can be used to determine the mot


and

L= T −U =12

m(lθ)2 + mgl cos θ.

mm

lθ

Fig. 7.1.2

Applying Theorem 7 .1.2, we know that

∂ ∂θ

[12

m(lθ)2 + mgl cos θ]− ddt

∂ ∂ θ

[12

m(lθ)2 + mgl

That is,

θ +gl

sin θ = 0 .

7.1.5 Hamiltonian Field. A Hamiltonian eld is a mdifferentiable Hamiltonian H : T M →R determined by



H (t) q(t) p (t)) =n

pi qi L(t q(t) q(t))

358 Chap

Theorem 7.1.3 Let T (M, ω) be a Hamiltonian eld with a HamiltoThen

dqi

dt=

∂H ∂p i

,dpi

dt= −

∂H ∂qi

for i = 1 , 2, · · ·, n .

Proof Consider the variation of S in (7-31). Notice that qiddpi . Applying (7-25), we know that

δS =

n

i=1 t2

t1[δpi dqi + pidδqi −

∂H ∂qi δqidt −

∂H ∂pi δpidt

Since

t2

t1

pidδqi = pi δqi|t2t1 − t2

t1

δqidpi

by integration of parts and δqi(t1) = δqi(t2) = 0, we nd that

t2

t1

pidδqi = − t2

t1

δqi dpi .

Substituting (7

−33) into (7

−32), we nally get that

δSn t2

[(d∂H

dt)δ (d +∂H

dt)δ ]



δS =i=1 t1

[(dqi−∂pidt)δpi−(dpi + ∂qi

dt)δqi].


By denition, the Lagrangian and Hamiltonian are relate

We can also directly deduce these Hamiltonian equations as fFor a xed time t, we know that

dL=n

i=1∂ L∂qi

dqi +n

i=1∂ L∂ qi

dqi .

Notice that

∂ L∂ qi

= pi and∂ L∂qi

= f i = ˙ pi

by (7 −20). Therefore,

dL=n

i=1

˙ pidqi +n

i=1

pidqi.


d(n

i=1

pi qi) =n

i=1

qidpi +n

i=1

pidqi .

Subtracting the equation (7 −35) from (7 −36), we get tha

d(

n

i=1 pi qi − L) =

n

i=1 qidpi −n

i=1 ˙ pidqi ,i.e.,



dH =n

qidpi

n

pidqi .

360 Chap

7.1.6 Conservation Law. A functional F (t, q(t), p (t)) on aT (M, ω) is conservative if it is invariable at all times, i.e., dF/dtshows that

dF dt

= ∂F ∂t

+n

i=1

( ∂F ∂qi

dqidt

+ ∂F ∂p i

dpidt

).

Substitute Hamiltonian equations into (7 −39). We nd that

dF

dt=

∂F

∂t+

n

i=1

(∂F

∂qi

∂H

∂p i −∂F

∂p i

∂H

∂qi).

Dene the Poisson bracket {H, F }of H, F to be

{H, F }P B =n

i=1

(∂F ∂qi

∂H ∂p i −

∂F ∂p i

∂H ∂qi

).

Then we have

dF dt

=∂F ∂t

+ {H, F }P B .

Theorem 7.1.4 Let T (M, ω) be a Hamiltonian mechanical eld. dqi

dt= {H, qi}P B ,

dpi

dt= {H, p i}P B



{ } { } for i = 1 , 2, · · ·, n .


dqi

dt= {H, qi}P B ,

dpi

dt= {H, p i}P B

for i = 1 , 2, · · ·, n .

If F is not self-evidently dependent on t, i.e., F = F ((7 −42) comes to be

dF dt

= {H, F }P B .

Therefore, F is conservative if and only if

{H, F

}P B = 0 in

if H is not self-evidently dependent on t, because of pi = ∂Lwe nd that

dH dt

=ddt

[n

i=1

pi qi − L(q(t), q(t))]

=n

i=1

( ˙ pi qi + pi qi ) −n

i=1

(∂ L∂qi

qi +∂∂

=n

i=1

( ˙ pi qi + pi qi ) −n

i=1

( ˙ pi qi + pi q

= 0 ,

i.e., H is conservative. Usually, H is called the mechanicT (M, ω), denoted by E . Whence, we have



Theorem 7.1.5 If the Hamiltonian H of a mechanical eid tl d d t t th T (M ) i ti f

362 Chap

where L (φ, ∂ μφ) is called the Lagrange density of eld. Applydensity, the Lagrange equations are generalized to the Euler-Lfollowing.

Theorem 7.1.6 Let φ(t, x ) be a eld with a Lagrangian Ldened

∂ μ∂ L

∂∂ μ φ −∂ L

∂φ= 0 .

Proof Now the action I is an integration of Lover time x0,

I =1

c d4xL (φ, ∂

μφ).

Whence, we know that

δI = d4x∂ L

∂φδφ +

∂ L

∂ (∂ μ φ)δ(∂ μ φ)

= d4x∂ L

∂φ −∂ μ∂ L

∂∂ μ φ δφ + ∂ μ∂ L

∂ (∂ μ φ) δφ

by the Hamiltonian principle. The last term can be turned into a over the boundary of region of this integration in which δφ = 0. Wintegral vanishes. We get that

δI = d4x ∂ L ∂φ −∂ μ ∂ L

∂∂ μ φδφ = 0

for arbitrary δφ. Therefore, we must have



∂∂ L ∂ L

= 0

Sec.7.2 Gravitational Field

which means the interaction take place instantly. Certainlycontradicted to the notion of modern physics, in which one asare carrying through intermediate particles. Even so, we wdiscussion at it since it is the fundamental of modern gravitat

The universal gravitational law of Newton determines themasses M and n of distance r to be

F = −GMm

r 2

with G = 6 .673×10− 8cm3/gs 2, which is the fundamental of N

eld. Let ρ(x) be the mass density of the Newtonian gravitx = ( x,y,z )∈R 3. Then its potential energy Φ(x) at x is de

Φ(x) = − Gρ(x )x −x

d3x .

Then

∂ Φ(x)∂x

= − ∂ [ Gρ (x )x− x ]∂x

d3x = − Gρ(x )(x −xx −x 3

Similarly,

∂ Φ(x)

∂y=

− Gρ(x )(y

−y )

x −x 3d3x =

−F

∂ Φ(x)∂z

= −Gρ(x )(z −z )

x x 3 d3x = −F



∂z x −x

Whence, the force acting on a particle with mass m is

364 Chap

i.e., the potential energy Φ( x) is a solution of the Poisson equatio

7.2.2 Einstein’s Spacetime. A Minkowskian spacetime is a square of line element

d2s = ημν dxμdxν = −c2dt2 + dx2 + dy2 + dz2

where c is the speed of light and ημν is the Minkowskian metrics

ημν =⎡⎢

⎢⎢⎢⎣

−1 0 0 00 1 0 00 0 1 00 0 0 1

⎤⎥

⎥⎥⎥⎦

.

For a particle moving in a gravitational eld, there are twoacting on it. One is the inertial force . Another is the gravitationany reference frame for the gravitational eld is selected by the obsshown in Section 7.1. Wether there are relation among them? Tby principles of equivalence and covariance following presented bafter a ten years speculation.

[Principle of Equivalence ] These gravitational forces and ineron a particle in a gravitational eld are equivalent and indistinguiother.

[Principle of Covariance ] A equation describing the law of pthe same form in all reference frame.



The Einstein’s spacetime is in fact a curved R 4 spacetime ( xRi i i h h f li l


|gμν | =

g00 g01 g02 g03

g10 g11 g12 g13

g20 g21 g22 g23

g30 g31 g32 g33

< 0.

For a given spacetime, let ( x0, x1, x2, x3) be its coordina

x μ = f μ (x0, x1, x2, x3)

another coordinate transformation, where μ = 0 , 1, 2 and 3.

g =∂x∂x

=

∂f 0

∂x 0 · · · ∂f 3

∂x 0

· · · · · · · · ·∂f 0

∂x 3 · · · ∂f 3

∂x 3

= 0 ,

then we can invert the coordinate transformation by

xμ = gμ (x 0, x 1, x 2, x 3),

and the differential of the two coordinate system are related

dxμ

=∂x μ

∂x ν dxν

=∂f μ

∂x ν dxν ,

dxμ =∂x μ

∂x ν dxν =∂gμ

∂x νdx ν .



∂x ∂x ν The principle of covariance means that gμν are tensors,

366 Chap

particularly, let T αβ be the metric tensor gμν , we get that

g =∂x∂x

2

g.

Besides, by calculus we have

d4x =∂x∂x

2

d4x.

Combining the equation (7

−45) with (7

−46), we get a relation fo

elements:

−g d4x = √−gd4x,

which means that the expression √−

gd4x is an invariant volume

7.2.3 Einstein Gravitational Field. By the discussion of Sgravitational eld equations should be constrained on principles ofcovariance, which will go over into the Poisson equation

∇

2

Φ(x) = 4πGρ (x),i.e., Newtonian eld equation in a certain limit, where

∂2 ∂2 ∂2



∇2 =

∂ 2

∂x+

∂ 2

∂y+

∂ 2

∂z.


The Einstein gravitational equations (7 −48) can be alsotonian principle. Choose the variational action of gravitationa

I =

√−

g(LG

−2κLF )d4x,

where LG = R is the Lagrangian for the gravitational led athe Lagrangian for all other elds with f ,α = ∂/∂x α for a energy-momentum tensor T μν to be

T μν

=2

√−g

∂ √−

gLF

∂gμν −∂

∂x α

∂ √−

gLF

∂gμν ,α

Then we have

Theorem 7.2.1 δI = 0 is equivalent to equations (7 −48)

Proof We prove that

δI = √−g(Rμν −12

gμν R −κT μν )δgμν d4x.

Varying the rst part of the integral (7 −49), we nd t

δ √−gRd4x = δ √−ggμν Rμν d4x

= √−ggμν δRμν d4x + Rμν δ(√−



Notice that

368 Chap

√−ggμν Rμν = √−ggμν ∂ (δΓρμν )

∂x ρ −∂ (δΓρ

μρ )∂x ν

= √−g∂ (gμν δΓρ

μν )∂x ρ −

∂ (gμν δΓρμρ )

∂x ν

= √−g∂ (gμν δΓαμν )

∂x α −∂ (gμα δΓρμρ )

∂x α

= √−g∇α V α ,

where V α = gμα δΓρμρ −gμα δΓρ

μρ is a contravariant vector and

∇α V α = ∂V α

∂x α + Γ αμα V μ ,

where

Γαμα = gαν Γνμα =

12g

∂g∂x ν =

1√−g

∂ √−g∂x ν .

Applying the Gauss theorem, we know that

√−ggμν δRμν d4x = ∂ (√−gV α )∂x α d4x = 0

for the rst integral on the right-hand side of (7 −51).Now the second integral on the right-hand side of (7 −51) g

Rμν δ(√−ggμν )d4x = √−gRμν δ(gμν )d4x + Rμν gμν δ(

√ R δ( μν )d4 Rδ(√ )



= √−gRμν δ(gμν )d4x + Rδ(√−g)


δ √−gRd 4x = √−g(Rμν −12

gμν R)δgμν d

for the variation of the gravitational part of the action (7 −LF (gμν , gμν ,α ) by assumption. For its second part, we obtain

δ √−gLF d4x = ∂ (√−gLF )∂gμν δgμν +

∂ (√−g∂gμ

,

The second term on the right-hand-side of the above equatiosurface integral which contributes nothing for its vanishing o

integration boundaries, minus another term following,

δ √−gLF d4x = ∂ (√−gLF )∂gμν δgμν −

∂ ∂x α

∂

=1

2 √−

gT μν δgμν d4x.

Summing up equations (7 −49), (7 −51), (7 −54) and that

δI =

√−g(Rμν −

12

gμν R −κT μν )δgμν

namely, the equation (7 −49). Since this equation is assuarbitrary variation δgμν , we therefore conclude that the integbe zero, i.e.,



1

370 Chap

Whence,

R00 = κT 00 +12

g00R

κT 00 +

1

2η00R =

1

2κT 00 =

1

2κc2

ρ(x),where ρ(x) is the mass density of the matter distribution.

Now by Theorem 5.3.4, we know that

Γk

00=

1

2gkλ 2

∂gλ0

∂x 0 −∂g00

∂x λ

−12

ηkλ ∂g00

∂x λ =12

δkl ∂g00

∂x l =12

∂g00

∂x k .

Therefore,

R00 = ∂ Γρ00

∂x ρ −∂ Γρ0ρ

∂x 0 + Γ σ00Γρ

ρσ −Γσ0ρΓρ

0σ

∂ Γρ00

∂x ρ∂ Γs

00

∂x s12

∂ 2g00

∂x s ∂x s =12∇

2g001c2∇

2

Equating the two expressions on R00 , we nally get that

∇2Φ(x) = 4πGρ (x),

where κ = 8πGc4 .

7 2 5 Schwarzschild Metric A Schwarzschild metric is a sph



7.2.5 Schwarzschild Metric. A Schwarzschild metric is a sphRi i i


d2s = B(r )dt2 −A(r )dr 2 −r 2dθ2 −r 2 sin2 θdφ

i.e., g00 = gtt = B(r ), g11 = grr = −A(r ), g22 = gθθ = −r 2,

and all other metric tensors equal to 0. Therefore, gtt = 1gθθ = −1/r 2 and gφφ = −1/r 2 sin2 θ.

For solving Einstein gravitational eld equations, we neezero connections Γρ

μν . By denition, we know that

Γρμν =gρσ

2∂g

σμ∂x ν +∂g

σν ∂x μ −∂g

μν ∂x σ .

Notice that all non-diagonal metric tensors equal to 0. C

Γrφφ = −

grr

2∂gφφ

∂x r = −12

(−1A

)∂ ∂

(r 2 sin2 θ) = −Similarly,

Γrrr =

A2A

, Γttt =

B2B

, Γtrr =

B2A

, Γθrθ = Γ φ

rφ =1r

Γrθθ = −

rA

, Γrφφ = −

rA

sin2 θ, Γφθφ = cot θ, Γθ

φφ =

where A = dAdr , B = dB

dr and all other connections are equal

Now we calculate non-zero Ricci tensors. By denition,



∂Γρ ∂Γρ

372 Chap

R11 = R rr = −∂

∂x r (Γ rrr + Γ θ

rθ + Γ φrφ + Γ t

rt ) −∂ Γr

rr

∂x r

+(Γ rrr Γr

rr + Γ θrθ Γθ

rθ + Γ φrφ Γφ

rφ + Γ trt Γt

rt )

−Γr

rr(Γ r

rr+ Γ θ

rθ+ Γ φ

rφ+ Γ t

rt)

=2r

+B2B

+2r 2 +

B 2

4B 2 −A2A

2r

=BB −B 2

2B 2 +B 2

4B 2 −A B4AB −

ArA

.

Similar calculations show that all Ricci tensors are as follows:

R tt = −B2A

+B4A

AA

+BB −

BrA

,

R rr =B2B −

B4B

AA

+BB −

ArA

,

Rθθ =r

2A −AA +

BB +

1A −1,

Rφφ = sin 2 θRθθ and Rμν = 0 if μ = ν.

Our object is to solve Einstein gravitational eld equations ii.e., Rμν = 0. Notice that

R tt

B+

Rrr

A= −

1rA

AA

+BB

= −BA + AB

rA 2B=

that is, BA + AB = ( AB ) = 0. Whence, AB =constant.N h i i i h li l (7 56)



Now at the innite point ∞, the line element (7 −56) s


Rθθ = rB + B −1 =ddr

(rB ) −1 = 0

Therefore, rB (r ) = r −r g, i.e., B (r ) = 1 −r g/r . Whenshould turn to at. In this case, Einstein gravitational eld e

Newtonian gravitational equation, i.e., r g = 2 Gm. Thereafte

B(r ) = 1 −2Gm

r.

Substituting (7 −61) into (7 −57), we get the Schwarzschil

ds2 = 1 −2mG

rdt2 −

dr 2

1 − 2mGr

−r 2dθ2 −r 2

or

ds2 = 1 −r g

rdt2 −

dr 2

1

−r g

r−r 2dθ2 −r 2 sin2 θd

We therefore obtain the covariant metric tensor for the sphericitational led following:

gμν =⎡

⎢⎢⎢⎢⎣

1 −r g

r 0 0 00 − 1 − r g

r− 1 0 0

0 0 −r 2 00 0 0 −r 2 sin2 θ

By (7 −63), we also know that the innitesimal distance of t



space is

374 Chap

Then there are 3 non-zero connections Γ ρμν more than (7-58) in th

Γrtr =

A2A

, Γttt =

B2B

, Γtrr =

A2B

,

where A = ∂A∂t and B = ∂B

∂t . These formulae (7

−59) are turned t

R rr =B2B −

B 2

4B 2 −A B4AB −

AAr

+A

2B −AB4B 2 −4

Rθθ = −1 +1A −

rA2A2 +

rB2AB

,

Rφφ = Rθθ sin2 θ,

R tt = −B2A

+A B4A2 −

BAr

+B 2

4AB+

A2A −

A2

4A2 −R tr = −

AAr

and all other Ricci tensors R rθ = Rrφ = Rθφ = Rθt = Rφt = 0.

Rμν = 0 implies that A = 0. Whence, A is independent on t. We

R rr =B2B −

B 2

4B 2 −A B4AB −

AAr

,

and

R tt = −B2A

+ A B4A2 − B

Ar+ B 2

4AB.

They are the same as in (7 −59). Similarly,



R R 1 A B


There is another way for solving Einstein gravitational a spherically symmetric distribution of matter, i.e., expressesand dr 2 in exponential forms following

ds2 = eν dt2

−eλ dr 2

−r 2(dθ2 + sin 2 θdφ

In this case, the metric tensors are as follows:

gμν =⎡⎢

⎢⎢⎢⎣

eν 0 0 00 −eλ 0 00 0 −r 2 0

0 0 0 −r 2 sin2 θ

⎤

⎦Then the nonzero connections are then given by

Γttt =

ν

2, Γt

tr =ν

2, Γt

rr =λ

2eλ− ν ;

Γrtt =

ν 2

eλ− ν , Γrtr =

λ2

, Γrrr =

λ2

;

Γrθθ = −re − λ , Γr

φφ = −r 2 sin2 θe− λ , Γ

Γθφφ = −sin θ cos θ, Γφ

rφ =1r

, Γφθφ = co

Then we can determine all nonzero Ricci tensors Rμν and nof equations Rμν = 0.

7.2.6 Schwarzschild Singularity. In the solution (7 −



important to the structure of Schwarzschild spacetime ( ct r θ

376 Chap

from timelike to spacelike. Whence, the two regions r > r s and rsmoothly at the surface r = r s .

We can also nd this fact if we examine the radical null direcφ = 0. In such a case, we have

ds2 = 1 − r s

rdt2 − 1 − r s

r− 1 dr 2 = 0 .

Therefore, the radical null directions must satisfy the following equ

drdt

= ± 1 −r s

r

in units in which the speed of light is unity. Notice that the timelicontained within the light cone, we know that in the region r > light cone decreases with r and tends to 0 at r = r s , such as those following.

¹

t

r s r

Fig. 7.2.1



In the region r < r the parametric lines of the time t becom


gμν =⎡⎢⎢

⎢⎢⎣

f 2 0 0 00 −f 2 0 00 0 −r 2 0

0 0 0 −r2

sin2

θ

⎤⎥⎥

⎥⎥⎦

.

Identifying (7 −63) with (7 −64), and requiring the functionand to remain nite and nonzero for u = v = 0, we nd a tthe exterior of the spherically singularity r > r s and the quavariables following:

v =rr s −1

12

expr

2r ssinh

t2r s

u =rr s −1

12

expr

2r scosh

t2r s

The inverse transformations are given by

rr s −1 exp

r2r s

= u2 −v2,

t2r s = arctanh

vu

and the function f is dened by

32G 3



32Gm 3 r

378 Chap

Notice that this transformation also presents a bridge between tclidean spaces in topology, which can be interpreted as the throconnecting two distant regions in a Euclidean space.

§7.3 ELECTROMAGNETIC FIELD

An electromagnetic eld is a physical eld produced by electricallIt affects the behavior of charged objects in the vicinity of the indenitely throughout space and describes the electromagnetic int

This eld can be viewed as a combination of an electric eleld. The electric eld is produced by stationary charges, and the mmoving charges, i.e., currents, which are often described as the sourmagnetic eld. Usually, the charges and currents interact with the eld is described by Maxwell’s equations and the Lorentz force la

7.3.1 Electrostatic Field. An electrostatic eld is a region of sby the existence of a force generated by electric charge. Denote byon an electrically charged particle with charge q located at x, duof a charge q located at x . Let ∇= ( ∂

∂x 1, ∂

∂x 2, ∂

∂x 3). According t

this force in vacuum is given by the expression

F (x) =qq

4πε 0

x −x

|x −x |3= −

qq4πε 0∇

1

|x −x |,

A vectorial electrostatic eld E stat is dened by a limiting pr



Sec.7.3 Electromagnetic Field

E stat (x) =q

4πε 0

x −x

|x −x |3= −

q4πε 0∇

1

|x −x |If there are m discrete electric charges qi located at

1, 2, 3,

· · ·, m, the assumption of linearity of vacuum allows u

individual electric elds into a total electric eld

E stat (x) =1

4πε 0

m

i=1

qx −xi

|x −xi|3.

Denote the electric charge density located at x withinwhich is measured in C/m 3 in SI units. Then the summationby an integration following:

E stat (x) =1

4πε 0

V

d3(x )ρ(x )x −x

|x

−x

|3

= −1

4πε 0 V d3(x )ρ(x )∇

1

|x −= −

14πε 0∇ V

d3(x )ρ(x )

|x −x |,

where we use the fact that ρ(x ) does not depend on the un

which ∇operates. Notice that under the assumption of lineequation (7-68) is valid for an arbitrary distribution of electdiscrete charges, in which case ρ can be expressed in the Dfollowing:



380 Chap

= −1

4πε 0 V d3(x )ρ(x )∇

2 1

|x −x |=

1ε0 V

d3(x )ρ(x )δ(x −x i) =ρ(x)ε0

.

Notice that

∇×(

∇

α(x)) = 0 for any scalar eld α(x), x

∈

R 3

get that

∇×E stat (x) = −1

4πε 0∇× ∇ V d3(x )

ρ(x )

|x −x |= 0 ,

which means that E stat is an irrotational eld. Whence, a electroscharacterized in terms of two equations following:

∇ ·E stat (x) = −ρ(x)ε0

,

∇×E stat (x) = 0 .

7.3.2 Magnetostatic Field. A magnetostatic eld is generacharge carriers such as electrons move through space or within an etor, and the interaction between these currents. Let F denote suca small loop C , with tangential line element dl located at x and I in the direction of dl, due to the presence of a small loop C welement dl located at x and carrying a current I in the directithose shown in Fig.7 .3.1.

C




F (x) =μ0II

4π C dl C

dl ×x −x

|x −x |3= −

μ0II 4π C

dl × C dl ×∇

1

|x

−where μ0 = 4π ×10− 7 ≈1.2566×10− 6H/m . Since a ×(b ×= ba ·c −ca ·b , we know that

F (x) = −μ0II

4π

C

dl

C

dl∇1

|x

−x

|−

μ0II 4π C C

Notice that the integrand in the rst integral is an exact differWe get that

F (x) = −μ0II

4π

C

C

x −x

|x

−x

|3 dldl

A static vectorial magnetic eld B stat is dened by

dB stat (x) =μ0I 4π

dl ×x −x

|x −x |3,

which means that dB stat at x is set up by the line eleme

magnetic ux density . Let dl = j(x )d3x . Then

B stat (x) =μ0

4π Vd3x j(x ) ×

x −x

|x −x



382 Chap

Since∇ ·(∇×a) = 0 for any a , we get that

∇ ·B stat (x) =μ0

4π∇ · ∇× V d3x

j(x )

|x −x |= 0 .

Applying

∇×(

∇×a) =

∇

(

∇ ·a)

−∇2a =

∇∇ ·a

−∇ ·∇a , we

∇×B stat (x) =μ0

4π∇× ∇× V d3x

j(x )

|x −x |= −

μ0

4π V d3x j(x )∇

2 1

|x −x |+

μ0

4π V d3x [ j(x ) ·∇

Notice that ∇ ·(αa) = a ·∇α + α∇ ·a . Integrating the second know that

V d3x [ j(x ) ·∇]∇

1

|x −x |= xk

V d3

x∇ j(x )

∂

∂x k

1

|x −x | − V d3

x [∇· j(x

= xk S d3x n j(x )

∂ ∂x k

1

|x −x | − V d3x [∇· j(x )]∇

where n is the normal unit vector of S directed along the outwar

x1 = sin θ cos φ

r + cos θ cos φ

θ + sin φ

φ,

x2 = sin θ sin φ r + cos θ sin φ θ + cos φ φ,

x3 = cos θ r −sin θ θ

and




Whence, a magnetostatic led can be characterized in termslowing:

∇ ·B stat (x) = 0 ,

∇×B stat (x) = μ0 j(x).

7.3.3 Electromagnetic Field. A electromagnetic led are dependent on both position x and time t. In this case, letdependent electric current density, particularly, it can be denewhere v is the velocity of the electric charge density ρ for simpcharge conservation law can be formulated in the equation of

∂ρ(t, x )∂t

+ ∇ · j(t, x ) = 0 ,

i.e., the time rate of change of electric charge ρ(t, x ) is bain the electric current density j(t, x ). Set ∇ · j(t, x ) = −∂ρ(derivation of equation (7

−77), we get that

∇×B (t, x ) = μ0 V d3x j(t, x )δ(x −x ) +

μ0

4π∂ ∂t V

d3x

= μ0 j(t, x ) + μ0∂ ∂t

ε0E (t, x ),

where

E (t, x ) = −1

4πε 0∇ V d3x

ρ(t, x )

|x −x |and it is assumed that



384 Chap

would be some energy loss unless the medium is superconducting. this energy is expended is j ·E per unit volume. If E is irrotation j will decay away with time. Stationary currents therefore requireeld which corresponds to an electromotive force (EMF) , denotedpresence of such a eld E EM F , the Ohm s law takes the form fol

j(t, x ) = σ(E stat + E EM F ),

where σ is the electric conductivity (S/m). Then the electromotivby

E = C dl ·(E stat + E EM F ),

where dl is a tangential line element of the closed loop C . By (7−70,which means that E stat is a conservative eld. This implies thintegral of E stat in above vanishes. Whence,

E = C dl ·E EM F .

Experimentally, a nonconservative EMF eld can be produced iC if the magnetic ux through C varies with time. In Fig.7 .3.2,

varying magnetic ux induced by a loop C which moves with velocvarying magnetic eld B(x).

d2xnB(x)B(x)




E (t) = C dl ·E (t, x ) = −

ddt

Φm (t)

= −ddt S

d2x

n ·B (t, x ) = − S

d2x

n ·

∂ ∂t

B

where Φm is the magnetic ux and S the surface encircled btheorem

C a ·dl = S

dS ·(∇×a)

in R 3 to (7

−82), we nd the differential equation following

∇×E (t, x ) = −∂ ∂t

B (t, x ).

Similarly, we can also get the following likewise that of

∇ ·B (t, x ) = 0 and ∇ ·E (t, x ) = 1ε0

ρ(x)

7.3.4 Maxwell Equation. All of (7−80), (7−83) and (7equations , i.e.,

∇ ·E (t, x ) = 1ε0

ρ(x),

∇×E (t, x ) = −∂ ∂t B (t, x ),

∇ ·B (t, x ) = 0 ,



386 Chap

of rank 2 called the electromagnetic eld tensor , where ∂ μ = (representation, the contravariant eld tensor can be written as follo

F μν =⎡

⎢⎢⎢⎢⎣

0 E x /c E y/c E z /cE x /c 0 Bz By

E y/c B z 0 Bx

E z /c B y Bx 0

⎤

⎥⎥⎥⎥⎦

.

Similarly, the covariant eld tensor is obtained from the cotensor in the usual manner by index lowering

F μν = gμκ gνλ F κλ = ∂ μ Aν −∂ ν Aμ

with a matrix representation

F μν =⎡

⎢⎢⎢⎢⎣

0 E x /c E y /c E z /c

−E x /c 0 −Bz By

−E y/c B z 0 −Bx

−E z /c −By Bx 0

⎤

⎥⎥⎥⎥⎦

.

Then the two Maxwell source equations can be written

∂ μ F μν = μ0 j ν . (

In fact, let ν = 0 corresponding to the rst/leftmost columrepresentation of the covariant component form of the electromagnF μν , we nd that



388 Chap

∂ μ∂ L EM

∂ (∂ μ Aν )=

14μ0

∂ μ∂

∂ (∂ μ Aν )(F κλ F κλ )

=1

4μ0∂ μ

∂ ∂ (∂ μ Aν )

[(∂ κ Aλ −∂ λ Aκ )(∂ κ Aλ

= 12μ0

∂ μ ∂ ∂ (∂ μ Aν )

(∂ κ Aλ ∂ κ Aλ −∂ κ Aλ ∂ λ A

But

∂

∂ (∂ μ Aν )(∂ κ Aλ ∂ κ Aλ ) = ∂ κ Aλ ∂

∂ (∂ μ Aν )∂ κ Aλ + ∂ κ Aλ

∂

∂ (∂ μ Aν

= ∂ κ Aλ ∂ ∂ (∂ μ Aν )

∂ κ Aλ + ∂ κ Aλ∂

∂ (∂ μ Aν

= ∂ κ Aλ ∂ ∂ (∂ μ Aν )

∂ κ Aλ + gκα gλβ ∂ κ Aλ

= ∂ κ Aλ ∂

∂ (∂ μ Aν )∂ κ Aλ + ∂ α Aβ ∂

∂ (∂ μ Aν= 2 ∂ μ Aν .

Similarly,

∂ ∂ (∂ μ Aν )

(∂ κ Aλ ∂ λ Aκ ) = 2∂ ν Aμ .

Whence,

∂ μ∂ L EM

∂ (∂ μ Aν )=

1μ0

∂ μ (∂ μ Aν −∂ ν Aμ ) =1μ0

∂ μ F μν




μνκλ =⎧⎪⎪⎨

⎪⎪⎩

1 if μνκλ is an even permutation o0 if at least two of μ,ν,κ,λ are eq

−1 if μνκλ is an odd permutation of

Then the dual electromagnetic tensor ∗F μν is dened by

∗F μν =12

μνκλ F κλ ,

or in a matrix form of the dual eld tensor following

∗F μν =⎡⎢⎢⎢⎢⎣

0 −cBx −cBy −cBzcBx 0 E z −E ycBy −E z 0 E xcBz E y −E x 0

Then the covariant form of the two Maxwell eld equati

∇×E (t, x ) = −∂ ∂t B (t, x ),

∇ ·B (t, x ) = 0

can then be written

∂ ∗

F μν

= 0 ,or equivalently,

∂ κ F μν + ∂ μ F νκ + ∂ ν F κμ = 0 ,



390 Chap

By (7 −66), we know that E (r ) = q/r 2 and

F μν =E (r )

c2

⎡⎢⎢

⎢⎢⎣

0 −1 0 01 0 0 00 0 0 0

0 0 0 0

⎤⎥⎥

⎥⎥⎦

and F μν =E (r )

c2

⎡⎢⎢

⎢⎢⎣

0

−1 0

0

i.e., F 01 = F 10 = E/c 2, F 10 = F 01 = −E/c 2 and all other entriescase, where indexes 0 = t, 1 = r, 2 = θ and 3 = φ. Calculations sh

F 01F 01 = F 10F 10 = −E 2/c 2,

F λτ F λτ = F 10F 01 + F 01F 10 = −2E 2.

In an electromagnetic led, we know that T μν = −(gσν F μdenition. Whence,

T 00 = −(g0σ F 0λ F σλ + E 22

g00) = E 22c4 B,

T 11 = −g11 (F 10F 10 +E 2

2) = −

E 2

2c4 A,

T 22 = E 2

2c4 r 2, T 33 = E 2

2c4 r 2 sin2 θ

and all of others T μν = 0, i.e.,

⎡B 0 0 0

⎤




R tt = −4Gπq2

c4r 4 B, R rr =4Gπq2

c4r 4 A

Rθθ = −4Gπq2

c4r 2 , Rφφ =4Gπq2

c4r 2 sin2

Similarly, we also know that

R tt

B+

R rr

A= 0 ,

which implies that A = 1 /B and

Rθθ =ddr (rB ) −1 = −

4Gπq2

c4r 2 .Integrating this equation, we nd that

rB −r =4Gπq2

c4r+ k.

Whence,

B(r ) = 1 +4Gπq2

c4r 2 +kr

.

Notice that if r → ∞, then

gtt = 1 −2Gmc2r

= 1 +4Gπq2

c4r 2 +kr

.

Whence k = −2Gm/c 2 and

B(r ) = 1 +4Gπq2

c4r 2 −2Gmc2r

.

C tl W t th t



392 Chap

§7.4 GAUGE FIELD

These symmetry transformations lies in the Einstein’s principle oflaws of physics should take the same form independently of anyare referred to as external symmetries . For knowing the behav

one also needs internal parameters, such as those of charge, baryetc., called gauge basis which uniquely determine the behavior of thunder consideration. The correspondent symmetry transformations oparameters, usually called gauge transformation , leaving invarianwhich are functional relations in internal parameters are termed in

A gauge eld is such a mathematical model with local or gunder a group, a nite-dimensional Lie group in most cases actbasis at an individual point in space and time, together with a set omaking physical predictions consistent with the symmetries of the mgeneralization of Einstein’s principle of covariance to that of internthe gauge theory can be applied to describe interaction of elementaperhaps, it maybe unies the existent four forces in physics. Uinvariance is adopted in a mathematical form following.

Gauge Invariant Principle A gauge eld equation, particuladensity of a gauge eld is invariant under gauge transformations o

7.4.1 Gauge Scalar Field. Let φ(x) be a complex scalar eThen its Lagrange density can be written as

L = ∂ μ φ†∂ μ φ −m2φ†φ,



Sec.7.4 Gauge Field

transformation. In this case, δφ = iγφ , δφ† = −iγφ †, δ∂

−iγ∂ μ φ†. Whence, we get that

δL = iγ ∂ L

∂φφ −φ†∂ L

∂φ† +∂ L

∂∂ μ φ∂ μφ

= iγ∂ μ∂ L

∂∂ μ φφ −φ† ∂ L

∂∂ μ φ†

by applying

∂ μ∂ L

∂ (∂ μφ) −

∂ L

∂φ= 0 , ∂ μ

∂ L

∂ (∂ μφ†) −

∂ L

∂φ†

Let δL = 0 in (7 −89), we get the continuous equation

∂ μ j μ = 0 ,

where

j μ =qi

∂ L

∂∂ μ φφ −φ† ∂ L

∂∂ μ φ† ,

i2 = −1 and q is a real number. Therefore,

j μ = iq(φ†∂ μ φ

−(∂ μ φ†)φ).

If γ is a function of x, i.e., γ (x), we need to nd the Lthis case. Notice that

∂ ( iγ ) iγ (∂ + i∂ )



394 Chap

L = ( Dμ φ)†(D μφ) −m2φ†φ.

Then we have

Dμ φ →(Dμφ) = Dμ φ = eiγ

Dμ φ,i.e., L is invariant under the transformation φ →φ = eiγ φ.

Now consider a set of n non-interacting complex scalar elds wm. Then an action is the sum of the usual action for each scalar following

I = d4xn

i=1

12

∂ μ φi∂ μ φi −12

m2φ2i .

Let Φ = ( φ1, φ2, · · ·, φn )t . In this case, the Lagrange density written as

L =12

(∂ μ Φ)t ∂ μ Φ −12

m2Φt Φ.

Then it is clear that the Lagrangian is invariant under the transformwhenever G is a n ×n matrix in orthogonal group O(n).

7.4.2 Maxwell Field. If a eld φ is gauge invariant in thφ(x) →φ (x) = eiγ (x )φ(x), then there must exists a coupling esuch that Aμ (x) is invariant under the gauge transformation

Aμ (x) →Aμ (x) = Aμ (x) + ∂ μ χ (x),



Sec.7.4 Gauge Field

By the denition of F μν and Jacobian identity establishthe following identity

∂ λ F μν + ∂ μ F νλ + ∂ ν F λμ = 0

holds. Whence, a Maxwell eld is determined by

∂ μ F μν = 0 ,

∂ κ F μν + ∂ μ F νκ + ∂ ν F κμ = 0 .

By the denition of F μν , the 4 coordinates used to descrcomplete independent. So we can choose additional gauge co

Lorentz Gauge: ∂ μ Aμ = 0.

Lorentz gauge condition is coinvariant, but it can not remfreedoms appeared in a Maxwell led. In fact, the number ofled is 3 after the Lorentz gauge added.

Coulomb Gauge: ∇ ·A = 0 and ∇2A0 = −ρ, where ρ

eld.

Radiation Gauge: ∇ ·A = 0 and A0 = 0 .

The Coulomb gauge and radiation gauge conditions rphysical freedoms in a Maxwell eld, but it will lose the fact, the number of freedom of a Maxwell led is 2 after radiation gauge added.

7.4.3 Weyl Field. A Weyl eld ψ(x) is determined by an



396 Chap

Let C = 0 and {bi , b j}= bib j + b j bi = −2gij . Then we obtain the d

∂ μ ∂ μ ψ = 0

from the equation (7 −90). Notice bi must be a matrix if bi b j + b

in a vector space with dimensional ≥2. For dimensional 2 space, bi = ±σi

where

σ1 =0 1

1 0

, σ2 =0 −i

i 0

, σ3 =1 0

0 −are Pauli matrixes and {σi , σ j}= −2gij . In this case, the Weyl be

∂ 0ψ = ±σi

∂ iψ.

Let

xi →xi = a i j x j

be a rotation transformation of the external eld of dimensional

is a 3 ×3 real orthogonal matrix with a ikak j = δi j . Correspondentransformation, let

ψ →ψ = Λψ



Sec.7.4 Gauge Field

We show the equation (7 −93) indeed has solutions. Corotation

a i j = gi

j + i jk θk .

of the external eld. Then its correspondent innitesimal ro

can be written as

Λ = 1 + iε iσi .

Substituting these two formulae into (7 −93) and neglectinmore than 2 of εi , we nd that

σi+ iε j (σ

iσ

j

−σ jσ

i) = σ

i+

i jk σ

jθ

k.

Solving this equation, we get that εi = θi / 2. Whence,

Λ = 1 −i2

θ ·σ,

where θ = ( θ1, θ2, θ3). Consequently, the Weyl equation is gaurotation of external eld if the internal eld rotates with ψ

The reection P and time-reversal transformation T oxi →a i

j x j , xi →bi j x j with (a i

j ), (bi j ) following

(aμν ) =⎡⎢⎢⎢⎢⎣

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

⎤⎥⎥⎥⎥⎦and (bμ

ν ) =⎡⎢⎢⎢⎢⎣

−1

000



398 Chap

acterizes a particle ψC with a reverse spiral of ψ. Whence, theinvariant under particle-antiparticle transformations C , but is inv

7.4.4 Dirac Field. The Dirac eld ψ(x) is determined by an eq

(iγ μ ∂ μ −m)ψ = 0 ,

where γ μ is a 4×4 matrix, called Dirac matrix and ψ a 4-compculation shows that

{γ μ , γ ν

}= γ μγ ν + γ ν γ μ = 2 gμν

and

γ 0 =I 2× 2 02× 2

02× 2 −I 2× 2, γ i =

02× 2 σi

−σi 02×

Now let

ψ =ψL

ψR,

where ψL , ψR are left-handed and right-handed Weyl spinors . Thetion can be rewritten as

(iγ μ ∂ μ −m)ψ = −m i(∂ 0 + σ ·∇)i(∂ 0 −σ ·∇) m

ψL

ψR

If we set m = 0, then the Dirac equation are decoupled to two Wey



Sec.7.4 Gauge Field

or equivalently,

Λγ μΛ− 1 = aμν γ ν .

Now let

Λ = I 4× 4 + 14

εμν γ μγ ν = 1 + 18

εμν (γ μγ ν −γ ν γ

where ενμ = −εμν . It can be veried that the identify (7 −9equation (7 −95) is covariant under the Lorentz transformati

Similar to the discussion of Weyl equation, we consider

equation under rotation, reection and time-reversal transform(1)Rotation. For an innitesimal rotation, εij = i

jk θk athem into (7 −97), we nd that

Λ = 1 −i2

θ ·Σ ,

where θ = (0 , θ1, θ2, θ3) and

Σ i = −i2

i jk γ j γ k =

σi 02× 2

02× 2 σi.

(2)Reection. Let xμ →aμν xν be a reection. Substitutin

haveΛ− 1γ 0Λ = γ 0, Λ− 1γ iΛ = γ i .

Solving these equations, we get that Λ = ηP γ 0, where ηP is a



400 Chap

Acting by Λ on the left side of (7−98), we get that

[i(Λγ 0Λ− 1∂ 0 −Λγ 1Λ− 1∂ 1 + Λγ 2Λ− 1∂ 2 −Λγ 3Λ− 1∂ 3) −m]Λψ∗

Comparing (7 −99) with (7 −95), we know that

Λγ 0Λ− 1 = γ 0, Λγ 1Λ− 1 =

−γ 1,

Λγ 2Λ− 1 = γ 2, Λγ 3Λ− 1 = −γ 3.

Solving these equations, we get that Λ = ηT γ 2γ 3, where ηT iη∗T ηT = 1. Whence, the time-reversal transformation of Dirac spinT ψ = ηT γ 2γ 3ψ∗.

(4)Particle-Antiparticle. A particle-antiparticle transformationis ψ →ψC = Cψ = iγ 2ψ∗. Assume spinor elds is gauge invariana gauge eld Aμ , the equation (7 −95) turns out

[γ μ (i∂ μ −qAμ ) −m]ψ = 0 ,

where the coupled number q is called charge. The complex conjuga(7 −100) is

[γ μ∗(−i∂ μ −qAμ ) −m]ψ∗= 0 .

Notice that Aμ is real and γ 2∗= −γ 2. Acting by iγ 2 on the equanally get that

[γ μ (i∂ μ + qAμ ) −m]ψC = 0 ,

Comparing the equation (7 −102) with (7 −100), we know th102) characterizes a Dirac eld of charge q Whence Dirac e



Sec.7.4 Gauge Field

ψ(x) →ψ (x) = e− iσ · θ ( x )

2 ψ(x),

where σ = ( σ1, σ2, σ3) are the Pauli matrices satisfying

[σi

2,

σ j

2] = iε ijk

σk

2, 1

≤i, j, k

≤and θ = ( θ1, θ2, θ3). For constructing a gauge-invariant Lagraduce the vector gauge elds A μ = ( A1

μ , A2μ , A3

μ ) to form cova

Dμ ψ = ∂ μ −igσ ·A μ

2ψ,

where g is the coupling constant. By gauge invariant principlsame transformation property as ψ, i.e.,

Dμ ψ →(Dμψ) = e− iσ · θ ( x )

2 Dμψ.

This implies that

∂ μ −ig σ ·A μ2

(e − iσ · θ (x

)2 ψ) = e − iσ · θ (x

)2 ∂ μ −ig

i.e.,

∂ μ e− iσ · θ ( x )

2 −igσ ·A μ

2e

− iσ · θ ( x )2 ψ = −ige

− iσ · θ2

Whence, we get thatσ ·A μ

2= e

− iσ · θ ( x )2

σ ·A μ

2e

iσ · θ ( x )2 −

ig

(∂ μ e− iσ · θ ( x )

2

which determines the transformation law for gauge elds. Foe



402 Chap

i.e.,

A iμ = Ai

μ + εijk θ j Akμ −

1g

∂ μ θi .

Similarly, consider the combination

(Dμ Dν −Dν Dμ )ψ = ig σi

2F iμν ψ

with

σ ·F μν

2= ∂ μ

σ ·A ν

2 −∂ ν σ ·A μ

2 −igσ ·A μ

2,

σ ·A2

i.e.,

F iμν = ∂ μ Aiν −∂ ν Ai

μ + gεijk A jμ Ak

ν .

By the gauge invariant principle, we have

[(Dμ Dν −Dν Dμ )ψ] = e− iσ · θ ( x )

2 (DμDν −Dν Dμ )ψ.

Substitute F iμν in (7 −103) into (7 −104), we know that

σ ·F μν e− iσ · θ ( x )

2 ψ = e− iσ · θ ( x )

2 σ ·F μν ψ,

i.e.,

σ ·F μν = e− iσ · θ ( x )

2 σ ·F μν eiσ · θ ( x )

2 .

For an innitesimal transformation θi 1, this translates int

F iμν = F iμν + εijk θ j F kμν .

Notice Fμν is not gauge invariant in this case. Whence, 14 Fμν



Sec.7.4 Gauge Field

F iμν = ∂ μ Aiν −∂ ν Ai

μ + gεijk A jμ Ak

ν ,

Dμψ = ∂ μ −igσ ·A μ

2.

Generally, the Lagrange density of Yang-Mills SU (n)-eld i

L = −12

T r(F μν a F aμν ).

Applying the Euler-Lagrange equations, we can also get the Yang-Mills SU (n) elds foe n ≥2.

7.4.6 Higgs Mechanism. The gauge invariance is in the ceeld theory. But it can be broken in adding certain non-Lagrangian by a spontaneous symmetry broken mechanism.

For example, let φ4 be a complex scalar eld with Lagr

L = ∂ μ φ†∂ μ φ −V (φ, φ†) = ∂ μ φ†∂ μ φ −m2φ†φ −λ

where m and λ are two parameters of φ. We have know thaunder the transformation

φ

→φ = eiγ φ

for a real number γ . Its ground state, i.e., the vacuum statewith minimal potential, namely,



404 Chap

i.e., |φ| = a. The equation (7 −105) has innite many solutionvacuum state is only one of them, i.e., the gauge symmetry is bno gauge symmetry in this case. Such eld is called Higgs eld . particle is called Higgs particle.

¹

·

V

−a a

Reφ

Imφ

Fig. 7.4.1

One can only observes the excitation on its average value a oiment. So we can write

φ(x) = a + 1√2(h(x) + iρ(x)) ,

where, by using the Dirac’s vector notation

v| = ( v1, v2, · · ·), |v = ( v1, v2, · · ·)t

and

v| · |u = ( v1, v2, · · ·) ·⎛⎜⎜⎝

u1

u2.⎞⎟⎟⎠

= v1u2 + v2u2 + · · ·=



Sec.7.4 Gauge Field

Abelian Gauge Field. Consider a complex scalar eld φis

L = ( ∂ μ −igAμ )φ†(∂ μ + igAμ )φ −m2φ†φ −λ(φ†φ)2 −= ∂ μ φ

†∂

μφ −m

2φ

†φ −λ(φ

†φ)

2

−igφ†

↔

∂ μ φAμ

+ g2φ

where Aμ is an Abelian gauge eld, F μν = ∂ μ Aν −∂ ν Aμ and

A↔∂ μ B = A

∂B∂x μ −

∂A∂x μ

with formulae following hold

A↔∂ μ (B + C ) = A

↔∂ μ B + A

↔∂ μ C,

(A + B)↔∂ μ C ) = A

↔∂ μ C + B

↔∂ μ C,

A↔∂ μ B = −B

↔∂ μ A,

A↔

∂ μ A = 0.

Choose the vacuum state φ in (7 −106) and neglect thave that

L =1

2(∂ μ h)2 +

1

2(∂ μ ρ)2

−λv2h2

−1

4F μν F μν +

− λvh (h2 + ρ2) −λ4

(h2 + ρ2)2 + gv∂ μρAμ

+ gh↔∂ μ ρAμ + g2vhAμAμ +

12

g2(h2 + ρ2)A



406 Chap

Whence, there are only gauge Aμ and Higgs, but without Goldstonunitary gauge eld.

Non-Abelian Gauge Field. Consider an isospin doublet ψ

eld under local SU (2) transformations. Its Lagrange density is

L = ( Dμ φ)†D μφ −m2φ†φ −λ(φ†φ)2 −14

F iμν F iμν

For m2 < 0, the vacuum state is in

0|φ†φ|0 = −m2

2λ= a2.

Now φ1 = χ 1 + iχ 2 and φ2 = χ 3 + iχ 4. Therefore,

φ†φ = χ 21 + χ 2

2 + χ 23 + χ 2

4,a sphere of radius a in he space of dimensional 4. Now we can cstate

φ(x) =1

√20

v + h(x) .

Calculations show that

V = m2φ†φ + λ(φ†φ)2 = λ(φ†φ)(φ†φ −v2) =λ4

((h2 + 2 v

(Dμφ)†D μ φ = ∂ μ φ†∂ μ φ + ig∂ μ φ†Aμ φ −igφ†Aμ ∂ μ φ + g

=12

(∂ μ h)2 +12

g2(v + h)2Aμ Aμ .



Sec.7.4 Gauge Field

7.4.7 Geometry of Gauge Field. Geometrically, a gaua choice of a local sections of principal bundle P (M, G ) and is a mapping between such sections. We establish such a mthis subsection.

Let P (M, G ) be a principal bre bundle over a manifold

by denition, there is a projection π : P →M and a Lie-groconditions following hold:

(1) G acts differentiably on P to the right without xP ×G →x ◦g∈P and x ◦g = x implies that g = 1 G ;

(2) The projection π : P

→M is differentiably onto a

{ p◦g|g∈G , π( p) = x}is a closed submanifold of P for x∈(3) For x ∈M , there is a local trivialization, also ca

T u of P over M , i.e., any x ∈M has a neighborhood U xT u : π− 1(U x ) →U x ×G with T u ( p) = ( π( p), su ( p)) such that

su

: π− 1(U x)

→G , s

u( pg) = s

u( p)g

for∀g∈G , p∈π− 1(U x ).

By denition, a principal bre bundle P (M, G ) is G -invit to be a gauge eld and nd its potential and strength in mthe connection 1-form, Ω = dω the curvature 2-form of a c

and s : M →P , π ◦s = id M be a local cross section of P (M

A = s∗ω =μ

Aμdxμ∈F 1(M 4),

F = s∗Ω = F dxμ∧dxν ∈F 2(M 4)



408 Chap

We explain the gauge elds discussed in this section are spemodel, particularly, the Maxwell and Yang-Mills SU (2) gauge etially mathematical meaning of spontaneous symmetry broken follo

Maxwell Gauge Field ψ. dimM = 4 and G = SO (2)

Notice that SO (2) is the group of rotations in the plane whivector v2 = v ·vt invariant. Any irreducible representation of equivalent to one of the unitary representation ϕn : S 1 →S 1 b

∀z∈S 1. In this case, any section of P (M,SO (2)) can be represens(ez) = z− n for e∈P , z∈S 1.

Consider the 1-form A as the local principal gauge potentiaconnection on a principal U (1)-bundle and the electromagnetic curvature. We have shown in Subsection 7 .3.4 that Maxwell eldequations ∂ μ F μν = μ0 j ν with the Jacobi identity. Let Ψ : M →Cof ψ by a section s : Ψ = ψs = s∗ψ. Then it is a gauge transforma

Yang-Mills Field. The Yang-Mills potentials Aα

= Aα

μ dxμ

givMills eld

B αμν =

∂Aαν

∂x μ −∂Aα

μ

∂x ν +12

cαρσ (Aρ

μAσν −Aρ

ν Aσμ ),

where cαρσ is determined in [X ρ , X σ] = cα

ρσ X α . Then

A2 = Aμ Aν dμdν =12

[Aμ , Aν ]dxμ dxν .

Now the gauge transformation in (7 −109) is



Sec.7.4 Gauge Field

We nally nd that

dA + A2 →dA + A 2 = U (dA + A2)U

i.e., F = dA + A2 is gauge invariant with local forms

F μν = ∂ μ Aν −∂ ν Aμ + [Aμ , Aν ],

which is just the F μν of the Yang-Mills elds by a proper ch

Spontaneous Symmetry Broken. Let Φ0 be the vacuumthe Lagrangian

L= L 1 + V (Φ), where V (Φ) stands for the i

a gauge group and g →ϕ(g) a representation of G . Dene

M 0 = ϕ(G )Φ0 = {ϕ(g)Φ0|g∈G }and G Φ0 = G 0 = {g∈

G |ϕ(g)Φ0 = Φ 0}is the isotropy subgM 0 is a homogenous space of G , i.e.,

M 0 = G / G 0 = {gG 0|g∈G }.

Denition 7.4.1 A gauge symmetry G associated with a Lagmodel Lis said to be spontaneously broken if and only if therM 0 dened in ( 7−108) obtained from a given vacuum state

If we require that V (Φ0) = 0 and V (ϕ(g)Φ) = V (Φ)Consequently, we can rewrite M 0 as

M 0 = {Φ|V (Φ) = 0}.



410 Chap

Physically, G 0 is important since it is the exact symmetry gi.e., the original gauge symmetry G is broken down to G 0 by Φ0.

For example, let L = L 1 + V (Φ) be an SO (3)-invariant LagV (Φ) = 1

2 μ2Φ2i − 1

4 λ(Φ2i )2, λ > 0. Then the necessary conditions

value of V (Φ) which characterizes spontaneous symmetry broken r

∂V ∂ Φi |Φi =Φ 0

i= 0 = μ2Φ2

i −λΦ2i Φi ⇒ Φ02

i =μ2

λWhence, the vacuum manifold M 0 of eld that minimize the potenby

M 0 = S 2 = Φi|Φ2i =μ2

λ ,

which corresponds to a spontaneous symmetry broken G = SO (By Denition 7.4.1, we know that

M 0 = SO (3)/SO (2)∼= S 2

on account of

ϕ(g)Φα0 = Φα

0 ⇔ϕ(g) =⎡⎢⎢⎣

0A 0

0 0 1

⎤⎥⎥⎦

, A∈SO

where Φα0 = (0 , 0, Φ0), Φ0 =

μ2/λ . Consequently, the natural C

SO (3) ×S 2 →S 2; (g, Φ) →ϕ(g)Φ; ϕ(g)Φ = Φ

is a transitive transformation.



Sec.7.5 Remarks

M is a manifold. Certainly, those of Weyl’s, Dirac’s, Maxwpartially differential equations discussed in this chapter are sucbehavior of elds usually reects geometrical properties with ithe dynamics behavior of elds. This fact enables us to detereld not dependent on the exact solutions of equations since

obtain, but on their differentially geometrical properties of mwe survey the gauge elds by principal bre bundles in Sethere are many such works should be carried out on this tren

7.5.2 Equation of Motion. The combination of the leasthe Lagrangian can be used both to the external and internal

determining the equations of motion of a eld. More techniqbe found in references [Ble1], [Car1], [ChL1], [Wan1], [Svethe quantum eld theory is essentially a theory established least action principle. Certainly, there are many works in thisboth in theoretical and practise, and nd the inner motivation

7.5.3 Gravitational Field. In Newtonian’s gravitational is transferred by eith and the action is at a distance, i.e., thinstantly. Einstein explained the gravitation to be concretea character of spacetime, not an external action. This meaRiemannian geometry in Einstein’s gravitational theory. Certa

ds deduces different structure of spacetime, such as those different metric we can nd. Which is proper for our WORLthe simplest metric, i.e., the Schwarzschild metric and its sonature Is it really happens so?



412 Chap

4-dimensional. But if we distinct the observed matter in a dimensioelectromagnetism, we do not even know weather the rest is still a dthe dimension 4 in electromagnetic theory is added by human beinits true color ?

7.5.5 Gauge Field with Interaction. Einstein’s principle of that a physical of external eld is independent on the articiallychosen by human beings. This is essentially a kind of symmetry A gauge symmetry is such a generalization for interaction. More resin references [Ble1], [ChL1], [Wan1] and [Sve1]. For its geometrreader is refereed to [Ble1]. Certainly, a gauge symmetry is depen

basis. Then how to choose its basis is a fundamental question. Wa concise ruler for all gauge elds? The theory of principal bresuch a tool. That is why we can generalize gauge symmetry to coin next chapter.

7.5.6 Unied Field. Many physicists, such as those of EinstVeblen, Pauli, Schouten and Thirty, · · ·etc. had attempted to conseld theory, i.e., the gravitational eld with quantum eld since have know an effective theory to unify the gravitational with electfor example, in references [Ble1], [Car1] and [Wes1]. By allowindimensional from 4 to 11, the String theory also presents a mathe

to unify the gravitational eld with quantum eld. In next chapter,their space structure by combinatorial differential geometry establis4−6 and show that we can establish innite many such unied eldcombinatorial notion in Section 2 .1. So the main objective for u



CHAPTER 8.

Combinatorial Fields with Applications

The combinatorial manifold can presents a naturally mathemcombinations of elds. This chapter presents a general ideai.e., how to establish such a model and how to determine its bmetrical properties or results on combinatorial manifolds . Fowe give a combinatorial model for elds with interactions in Swe determine the equations of elds in Section 8 .2, which arconsequence of Euler-Lagrange equations. It should be notof equations of combinatorial eld is dependent on the Lagra

with that of elds M i for integers 1 ≤ i ≤m, in which eachdetermine a geometry of combinatorial elds. Notice the sphsolution of Einstein’s eld equations in vacuum is well-knowthe line element ds of combinatorial gravitational elds in Sis not difficult if all these line elements ds i , 1 ≤ i ≤m are k

By considering the gauge bases and their combination, we iniwhat conditions on gauge bases can bring a combinatorial gaug8.4. We also give a way for determining Lagrange density by on surfaces, which includes Z2 gauge theory as its conclusion



414 Chap.8 Combinatorial Fiel

§8.1 COMBINATORIAL FIELDS

8.1.1 Combinatorial Field. The multi-laterality of WORLD imnatorial elds in discussion. A combinatorial eld C consists of with interactions between C i and C j for some integers i, j, 1 ≤combinatorial elds are shown in Fig.8 .1.1, where in (a) each paiteraction for integers 1 ≤ i = j ≤4, but in (b) only pair {C i , C i+1

i, i + 1 ≡(mod4).

C 4

C 1C 2 C 3

C 1

CC 4

C 3

(a) (b)

Fig. 8.1.1

Such combinatorial elds with interactions are widely existinlet C 1, C 2, · · ·, C n be n electric elds E stat

1 , E stat2 , · · ·, E stat

n . Then combinatorial eld with interactions. A combinatorial eld C nmulti-action S . For example, let F E i E j be the force action betw

immediately get a multi-action S = ∪i,j F E i E j between E stat1 , E s2

two multi-actions induced by combinatorial elds in Fig.8 .1.1 are

C 1 C 2¹

C 1 C¹



Sec.8.1 Combinatorial Fields

Notice that an action −→A always appears with an anti-acsuch a pair of action can be denoted by an edge −→A←−A . This fa vertex-edge labeled graph GL [C ] for a combinatorial eld

V (GL [C ]) = {v1, v2, · · ·, vn},

E (GL [C ]) = {viv j | if ∃−→C i←−C j between C i an

with labels

θL (vi) = C i , θL (viv j ) = −→C i←−C j .

For example, the vertex-edge labeled graphs correspondent t

in Fig.8.1.1 are shown in Fig.8.1.3, in where the vertex-edg(b) are respectively correspondent to the combinatorial eld (

C 1

C 2

C 3C 4

C 1

C 3

(a)

−→C 1←−C 2−→C 1←−C 3−→C 1←−C 4

−→C 2←−C 3−→C 2←−C 4

−→C 3←−C 4

−→C

−→C 4←−C 1

Fig. 8.1.3

We have know that a eld maybe changes dependent on




for an integer k ≥1. Generally, if dim(C 1 ∩C 2) = 0, there are noC 1 and C 2, and one can only observes 3-dimensional actions. Sresearch the structure of conguration space for combinatorial eld

8.1.2 Combinatorial Conguration Space. As we have sha eld can be presented by its a state function ψ(x) in a referencharacterized by partially differential equations, such as those of th

Scalar eld: (∂ 2 + m2)ψ = 0,

Weyl eld: ∂ 0ψ = ±σi∂ i ψ,

Dirac eld: (iγ μ

∂ μ −m)ψ = 0,These conguration spaces are all the Minkowskian. Then w

combinatorial elds ? Considering the character of elds, a natural terize each eld C i , 1 ≤ i ≤n of them by itself reference frame {we get a combinatorial conguration space, i.e., a combinatorial Eu

combinatorial elds. This enables us to classify combinatorial eldsinto two categories:

Type I. n = 1 .

In this category, the external actions between elds are all thtablish principles following on actions between elds.

Action Principle of Fields. There are always exist an actio elds C 1 and C 2 of a Type I combinatorial eld, i.e., dim(C 1 ∩C

Innite Principle of Action An action between two elds in




dimmin C = 4 + s,

where the integer s ≥0 is determined by

r + s −1

r< n

≤r + s

r.

Now if we can establish a time parameter t for all eTheorems 4 .1.2 and 4.1.5 imply the maximum dimension 2 ndimension

dimmin C =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

4, if n = 1 ,

5, if 2≤n ≤46, if 5≤n ≤13 + √n , if n ≥11.

of C (t). In this case, the action on a eld C i comes from i, 1 ≤ j ≤n in a combinatorial eld C (t), which can be dep

C i

C 1

C 2

C i− 1C i+1

C n

Ê

¹

¹

©

Á




a Type I combinatorial free-eld C (t) of scalar elds, Weyl eC 1, C 2, · · ·, C n by partially differential equation system as follows:

Combinatorial Scalar Free-Fields :

(∂ 2 + m21)ψ(ict,x 11 , x12, x13) = 0,

(∂ 2 + m22)ψ(ict,x 21 , x22, x23) = 0,

· · · · · · · · · · · · · · · · · · · · · · · · · · ·,

(∂ 2 + m2n )ψ(ict, x n 1, xn 2, xn 3) = 0.

Combinatorial Weyl Free-Field :

∂ 0ψ(ict,x 11 , x12, x13) = ±σi ∂ i ψ(ict,x 11 , x12 , x13),

∂ 0ψ(ict,x 21 , x22, x23) = ±σi ∂ i ψ(ict,x 21 , x22 , x23),

· · · · · · · · · · · · · · · · · · · · · · · · · · ·,∂ 0ψ(ict,x n 1, xn 2, xn 3) = ±σi ∂ iψ(ict,x n 1, xn 2, xn 3).

Combinatorial Dirac Free-Field :

(iγ μ ∂ μ −m1)ψ(ict,x 11 , x12 , x13) = 0,

(iγ μ ∂ μ −m2)ψ(ict,x 21 , x22 , x23) = 0,

· · · · · · · · · · · · · · · · · · · · · · · · · · ·,

(iγ μ ∂ μ −mn )ψ(ict, x n 1, xn 2, xn 3) = 0.

Type II. n ≥2.

In this category the external actions between elds are multi




(∂ 2 + m22)ψ(ict,x 21, x22 , x23) = 0,

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·,

(∂ 2 + m2k)ψ(ict, x k1, xk2, xk3) = 0.

∂ 0ψ(ict, x (k+1)1 , x (k+1)2 , x (k+1)3 ) = ±σi∂ i ψ(ict,x (k+

∂ 0ψ(ict, x (k+2)1 , x (k+2)2 , x (k+2)3 ) = ±σi∂ i ψ(ict,x (k+

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·∂ 0ψ(ict, x l1, x l2, x l3) = ±σi ∂ i ψ(ict,x l1, x l2, x l3).

(iγ μ ∂ μ −m l+1 )ψ(ict,x (l+1)1 , x (l+1)2 , x(l+1)3 ) = 0,

(iγ μ ∂ μ −m l+2 )ψ(ict,x (l+2)1 , x (l+2)2 , x(l+2)3 ) = 0,

· · · · · · · · · · · ·· · · · · · · · · · · ·· · · · · · · · · · ·,

(iγ μ ∂ μ −mn )ψ(ict,x n 1, xn 2, xn 3) = 0.

In this combinatorial led, there are respective complete and K

n − l+1in its underlying graph GL [C (t)].

8.1.3 Geometry on Combinatorial Field. In the vcan only observe behavior of particles in the eld where winformation in a combinatorial reference frame. So it is imgeometrical model for combinatorial elds.

Notice that each conguration space in last subsection is manifold. This fact enables us to introduce a geometrical mmanifold for a combinatorial eld C (t) following:




ω =μ,ν

ωμν dxμν = ωμν dxμν .

This model establishes the the dynamics on a combinatorial eus to apply results in Chapters 4 −6, i.e., combinatorial differe

characterizing the behaviors of combinatorial elds, such as thosT rs (M ), k-forms Λk(M ), exterior differentiation d : Λ(M ) →ΛD, Lie multi-groups L G and principle bre bundles P (M, L G ),binatorial Riemannian manifolds. Whence, we can apply the Einsprinciple to construct equations of combinatorial manifolds, i.e., tenits correspondent combinatorial manifold M of a combinatorial maybe any connected graph.

For example, we have known the interaction equations of grMaxwell eld and Yang-Mills eld are as follows:

Gravitational eld: Rμν − 12 gμν R = κT μν ,

Maxwell eld: ∂ μ F μν = 0 and ∂ κ F μν + ∂ μ F νκ + ∂ ν F

Yang-Mills eld: D μF aμν = 0 and Dκ F aμν + Dμ F aνκ + D

Whence, we can characterize the behavior of combinatorial of connection, curvature tensors, metric tensors, · · ·with a form f

F (κ1 λ 1 )··· (κ r λ r )(μ1 ν 1 )··· (μ s ν s ) = 0 .

Notice we can only observe behavior of particles in R 4 in practicthe tensor equation




book TAO TEH KING by words that all things we can ackby our eyes, or ears, or nose, or tongue, or body or passionsIn other words, with the help of developing technology, wrecognized scope. This recognizing process is endless. So athe WORLD with a proper precision is enough for various a

beings.

Strong Anthropic Principle The born of life is essencharacterization of WORLD at sometimes.

This principle means that the born of human beings is nevitable in the WORLD. Whence, there is a deep regulation ofthe human being come into being. In other words, one can nthen nally recognizes the whole WORLD, i.e., life appeaa denite conclusion of this regulation. So one wishes to mathematics, for instance the Theory of Everything .

It should be noted that one can only observes unilateral re

alluded also in the mortal of the proverb of six blind men andeach observation is meaningful only in a particular referencstein’s general relativity theory essentially means that a physicon the reference frame adopted by a researcher. That is whrial tensor equations to characterize a physical law in a co

determining the behavior of combinatorial elds, we need tfollowing, which is an extension of Einstein’s covariance prinelds.

P j i P i i l A h i l i bi i l




for integers 1 ≤μ i ≤s( p), 1 ≤ν i ≤nμ i with 1 ≤ i ≤s and 1 ≤κnκ j with 1 ≤ j ≤r , then for any integer μ, 1 ≤μ ≤s( p), there m

F (μλ 1 )··· (μλ r )(μν 1 )··· (μν s ) = 0

for integers ν i , 1 ≤ν i ≤nμ with 1 ≤ i ≤s.

Applying this projective principle enables us to nd solutiorial tensor equation characterizing a combinatorial eld underlyingstructure G in follows sections.

§8.2 EQUATION OF COMBINATORIAL FIELD

8.2.1 Lagrangian on Combinatorial Field. For establishequations of a combinatorial eld C (t), we need to determine its Lrst. Generally, this Lagrange density can be constructed by applyiits vertex-edge labeled graph GL [C (t)] for our objective. Applyiwe can formally present this problem following.

Problem 8.2.1 Let GL [M ] be a vertex-edge labeled graph of a comM consisting of n manifolds M 1, M 2, · · ·, M n with labels

θL : V (GL [M ])

→ {L M i , 1

≤i

≤n

},

θL : E (GL [M ]) → {T ij for ∀(M i , M j )∈E (GL [M ]

where L M i : T M i →R , T ij : T (M i ∩M j ) →R . Construct a func



Sec.8.2 Equation of Combinatorial Field

Case 1. Linear

In this case, the general expression of the Lagrange den

L GL [M ] =

n

i=1a i L M i +

(M i ,M j )∈E (GL [M ])bij T i

where a i , bij and C are undetermined coefficients in R . CL |M i of L

GL [M ] on M i , 1 ≤ i ≤n. We get that

L GL [M ]|M i = a i L M i +


Let a i = 1 and bij = 1 for 1 ≤ i, j ≤n and

L i

int = L M i , L i

ext = (M i ,M j )∈E (GL [M ])T

Then we know that

L GL [M ]|M i = L i

int + L iext ,

i.e., the projection L |M i of L GL [M ] on eld M i consists o

comes from the interaction L i in eld M i and the second caction L i

ext from elds M j to M i for ∀(M i , M j ) ∈E (GL [that external actions L i

ext between elds M i , M j for ∀(Mtransferred to an interaction of the combinatorial eld C (t)

If we choose a i = 1 but bij = −1 for 1≤ i, j ≤n, then

L GL [M ] =n

i=1L M i −(M i ,M j )∈E (GL [M ])

T ij

with its projection

i i




is connected in Chapter 4, which means that E (GL [M ]) = ∅onldo not choose this formula to be the Lagrange density of combinatdiscussion following.

Case 2. Non-Linear

In this case, the Lagrange density L G

L[M ]

is a non-linear funT ij for 1 ≤ i, j ≤ n. Let the minimum and maximum indexesE (GL [M ]) are il and iu , respectively. Denote by

x = ( x1, x2, · · ·) = ( L M 1 , L M 2 , · · ·, L M n , T 11 l , · · ·, T 11u , · ·If L

GL [M ] is k + 1 differentiable, k ≥0 by Taylor’s formula we kn

L GL [M ] = L

GL [M ](0) +n

i=1

∂ L GL [M ]

∂x i x i =0xi +

12!

n

i,j =1

∂ 2L G

∂x i

+ · · ·+1k!

n

i1 ,i 2 ,··· ,i k =1

∂ k L GL [M ]

∂x i1 ∂x i2

· · ·∂x ik

x i j =0 ,1≤ j ≤ k

x

+ R(x1, x2, · · ·),where

limx →0

R(x1, x2, · · ·)x

= 0 .

Certainly, we can choose the rst s terms

n ∂L GL [M ] 1 n ∂2L

GL [M




in most cases on combinatorial elds.Now we consider the net value of Lagrange density on

without intersections. Certainly, we can determine it by apexclusion principle. For example, if GL [M ] is K 3-free, similalary 4.2.4, we know that the net Lagrange density is

L GL [M ] =

(M i ,M j )∈E (GL [M ])

(L M i + L M j −T ij )

=(M i ,M j )∈E (GL [M ])

(L M i + L M j ) −(M i ,M j

=M i )∈V (GL [M ])

L 2M i −(M i ,M j )∈E (GL [M ])

T

which is a polynomial of degree 2 with a projection

L GL [M ]|M i = L 2

M i −(M i ,M j )∈E (GL [M ])

T

on the eld M i .Similarly, we also do not choose the expression

L s1M 1 + L s 2

M 2 + · · ·+ L s nM n

with s i ≥2 for 1≤ i ≤n to be the Lagrange density L GL [M ]

only if there are no actions between elds M i , M j for any inE (GL [M ]) =∅since it has physical meaning only if n = 1.




Likewise the Lagrange density, we can also determine the eqφ(x) by Hamilton density such as those of equations in Theorem 7

ddt

∂ L

∂φ= −

∂ H∂φ

,dφdt

=∂ H

∂ ∂ L

∂φ

.

Whence, for determining the equations of motion of a combinatorenough to nd its Hamilton density. Now the disguise of Problem the following.

Problem 8.2.2 Let GL [M ] be a vertex-edge labeled graph of a comM consisting of n manifolds M 1, M 2, · · ·, M n with labels

θL : V (GL [M ]) → {HM i , 1 ≤ i ≤n},

θL : E (GL [M ]) → {Hij for ∀(M i , M j )∈E (GL [M ]

where HM i : T M i →R , Hij : T (M i ∩M j ) →R . Construct a func

HGL [M ] : GL [M ]

→R

such that GL [M ] is invariant under the projection of HGL [M ] on

For elds M i , M i ∩M j , 1 ≤ i, j ≤n, we have known their Hto be respective

HM i = πi φM i −L M i and Hij = πij φM i ∩M j −T ij

by denition, where πi = ∂ L M i /∂ φM i and πij = ∂ T ij /∂ φM i ∩M j . Sof Lagrange densities we classify these Hamilton densities on line




=n

i=1

a iπi φM i +(M i ,M j )∈E (GL [M ])

bij πij φM i ∩M j

−n

i=1

a i L M i −(M i ,M j )∈E (GL [M ])

bij T ij + C

Similarly, let the minimum and maximum indexes j foare il and iu , respectively. Denote by

φ = ( a1φM 1 , · · ·, an φM n , b11 l φM 1 ∩M 1 l , · · ·, b11u φM 1 ∩M

π = ( π1, π2, · · ·, πn , π11 l , · · ·, π11u , · · ·, πnn u ).Then

φ, π =n

i=1

a iπi φM i +(M i ,M j )∈E (GL [M ])

bij πij

Choose a linear Lagrange density of the vertex-edge labeled

L GL [M ] = n

i=1a i L M i +


We nally get that

HGL [M ] = φ, π −L GL [M ],

which is a generalization of the relation of Hamilton densitydensity of a eld. Furthermore, if {HM i , Hij ; 1 ≤ i, j ≤n}ann}are orthogonal in this case, then we get the following con

Theorem 8 2 1 If the Hamilton density H is linear an




HGL [M ] = HGL [M ](HM i , Hij ; 1 ≤ i, j ≤n)

= HGL [M ](πi φM i −L M i , π ij φM i ∩M j −T ij ; 1 ≤Denote by

y = ( y1, y2, · · ·) = ( HM 1 , HM 2 , · · ·, HM n , H11 , · · ·, H11 l , · · ·, HIf HGL [M ] is s + 1 differentiable, s ≥0, by Taylor’s formula we kn

HGL [M ] = HGL [M ](0) +n

i=1

∂ HGL [M ]

∂yi yi =0yi +

12!

n

i,j =1

∂ 2HG

∂yi

+ · · ·+1s!

n

i1 ,i 2 ,··· ,i s =1

∂ sHGL [M ]

∂yi1 ∂yi2 · · ·∂yis yi j =0 ,1≤ j ≤ syi

+ K (y1, y2,

· · ·),

where

limy →0

K (y1, y2, · · ·)y

= 0 .

Certainly, we can also choose the rst s terms

HGL [M ](0) +n

i=1

∂ HGL [M ]

∂yi yi =0yi +

12!

n

i,j =1

∂ 2HGL [M

∂yi ∂y j




Φ = (A1φM 1 , · · ·, An φM n , B11 l φM 1 ∩M 1 l , · · ·, B11u φM 1 ∩M

π = ( π1, π2, · · ·, πn , π11 l , · · ·, π11u , · · ·, πnn u )and

L LGL [M ] = −HGL [M ](0) +

n

i=1

Ai L M i +(M i ,M j )∈E (G

where

Ai = (∂ HGL [M ]

∂ HM i H M i =0, Bij =

∂ HGL [M

∂ Hij

for 1

≤i, j

≤n. Applying formulae in (8

−1), we know tha

HLGL [M ] = HGL [M ](0) +

n

i=1

AiHM i +(M i ,M j )∈E (G

= HGL [M ](0) +n

i=1

Ai (πiφM i −L M i )

+(M i ,M j )∈E (GL [M ])

B ij (πij φM i ∩M j −T

= Φ, π −L LGL [M ].

That is,

HLGL [M ] = Φ, π −

L LGL [M ],

i.e., a generalization of the relation of Hamilton density wGenerally, there are no relation (8 −3) for the non-liner par

H with that of Lagrange density L

430




Combinatorial Scalar Fields.For a scalar eld φ(x), we have known its Lagrange density i

L =12

(∂ μφ∂ μ φ −m2φ2).

Now if elds M 1, M 2,

· · ·, M n are harmonizing, i.e., we can establis

φM on a reference frame {ict,x 1, x2, x3}for the combinatorial ecan choose the Lagrange density L

GL [M ] to be

L GL [M ] =

12

(∂ μ φM ∂ μ φM −m2φ2M ).

Applying (8

−4), we know that its equation is

∂ μ∂ L

GL [M ]

∂∂ μ φM −∂ L

GL [M ]

∂φM = ∂ μ ∂ μ φM + m2φM = ( ∂ 2 + m2

which is the same as that of scalar elds. But in general, M 1, Mharmonizing. So we can only nd the equation of M (t) by combin

Without loss of generality, let

φM =n

i=1

ci φM i ,

L GL [M ] = 1

2

n

i=1

(∂ μ i φM i ∂ μ i φM i −m2i φ2

M i ) +(M i ,M j )∈E (GL [M ])

bij

i.e.,

S 8 2 E i f C bi i l Fi ld




∂LGL [M ]

∂φM = −

n

i=1

m2i

ci+

(M i ,M j )∈E (GL [M ])

bijφM

ci

Whence, by (8 −4) we get the equation of combinatorial scal

n

i=1

1ci

(∂ μ ∂ μ i + m2i )φM i −

(M i ,M j )∈E (GL [M ])

bij φM j

ci+

This equation contains all cases discussed before.

Case 1. |V (GL [M ])| = 1

In this case, bij = 0 , ci = 1 and ∂ μ i = ∂ μ . We get the following

(∂ 2 + m2)φM = 0 ,

where φM is in fact a wave function of eld.

Case 2. Free

In this case, bij = 0, i.e., there are no action between CWe get the equation

n

i=1

(∂ μ ∂ μ i + m2i )φM i = 0 .

Applying the projective principle, we get the equations of comeld following, which is the same as in Section 8.1.2.

432 Chap 8 Combinatorial Fiel




n

i=1

(∂ μ ∂ μ i + m2i )φM i −

(M i ,M j )∈E (GL [M ])

bij (φM j + φM i )

Applying the projective principle again, we get the equations of comeld with interactions following.

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(∂ 21 + m21 −

(M 1 ,M j )∈E (GL [M ])b1 j )φM 1 = 0

(∂ 22 + m22 −

(M 2 ,M j )∈E (GL [M ])b2 j )φM 2 = 0

· · · · · · · · · · · · · · ·· · · · · · · · · · · · · · ·(∂ 2n + m2

n

−(M n ,M j )∈E (GL [M ])

bnj )φM n = 0 .

where, for an integer i, 1 ≤ i ≤n,(M i ,M j )∈E (GL [M ])

bij φM i is a ter

of elds M j to M i for any integer j such that ( M i , M j )∈E (GL

differential equation system can be used to determine the behavior scalar elds. Certainly, we can also apply non-linear action termbehavior and nd more efficient results on combinatorial scalar e

Combinatorial Dirac Fields.

For a Dirac eld φ(x), we have known its Lagrange density i

L = ψ(iγ μ ∂ μ −m)ψ.For simplicity, we consider the linear Lagrange density L

GL [M

for 1 ≤ i, j ≤n, i.e.,

Sec 8 2 Equation of Combinatorial Field




Let ci = 1 , 1 ≤ i ≤ n. Applying the projective princfollowing

⎧⎪⎪⎪

⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(iγ μ1 ∂ μ1 −m1 −(M 1 ,M j )∈E (GL [M ])

b1 j )ψM 1

(iγ μ2 ∂ μ2

−m2

−(M 2 ,M j )∈E (GL [M ])

b2 j )ψM 2

· · · · · · · · · · · ·· · · · · · · · · · · ·· · · ·(iγ μn ∂ μn −mn −

(M n ,M j )∈E (GL [M ])bnj )ψM

Certainly, if GL [M ] is trivial, we get the Dirac equati

to the discussion of combinatorial scalar elds, we can apply(8 −7) to determine the behavior of combinatorial Dirac eld

Combinatorial Scalar and Dirac Field.

Let C (t) be a combinatorial eld M of k scalar elds Dirac elds M 1, M 2,

· · ·, M s with GL [M ] = GL

S + GLD , w

respective induced subgraphs of scalar elds or Dirac eldsthe Lagrange density of C (t) to be a linear combination

L GL [M ] = L 1 + L 2 + L 3 + L 4 + L 5, and φM =

k

i=1

where L 1 = 12

n

i=1(∂ μ i φM i ∂ μ i φM i −m2

i φ2M i ),

n




434 C ap.8 Co b ato a e

k

i=1

1ci

(∂ μ ∂ μ i + m2i )φM i +

s

j =1

1c j (iγ μ i ∂ μ −m j )ψM j

−(M i ,M j )∈E (GL

S )

b1ij

φM j

ci+

φM i

c j −(M i ,M j )∈E (GL

D )

b2ij

ψ

−(M i ,M j )∈E (GL

S ,G LD )

b3ij

ψM j

ci+ φM i

c j ) = 0 .

For simplicity, let ci = c j = 1, 1 ≤ i ≤ k, 1 ≤ j ≤ s. Tprojective principle on a scalar eld M i , we get that

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(∂ 21 + m21 −(M 1 ,M j )∈E (GL

S )b1

1 j −(M 1 ,M j )∈E (GLS ,G L

D )b3

1 j )φM 1

(∂ 22 + m22 −(M 2 ,M j )∈E (GL

S )b1

2 j −(M 2 ,M j )∈E (GLS ,G L

D )b3

2 j )φM 2

· · · · · · · · · · · ·· · · · · · · · · · · ·· · · · · · · · · · · ·(∂ 2k + m2

k

−(M k ,M j )∈E (GLS )

b1kj

−(M k ,M j )∈E (GLS ,G LD )

b3kj )φM k

Applying the projective principle on a Dirac eld M j , we get tha

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(iγ μ1 ∂ μ1 −m1 −(M 1 ,M j )∈E (GLD )

b21 j −(M 1 ,M j )∈E (GL

S ,G LD )

b31 j )ψ

(iγ μ2 ∂ μ2 −m2 −(M 2 ,M j )∈E (GLD )

b22 j −(M 2 ,M j )∈E (GLS ,G L

D )b32 j )ψ

· · · · · · · · · · · · · · ·· · · · · · · · · · · · · · ·· · · · · ·(iγ μ s ∂ μ s −ms − b2

sj − b3sj )ψ




q

are linear action terms. We can use (8 −8) and (8 −9) to of combinatorial scalar and Dirac elds.

8.2.4 Tensor Equation on Combinatorial Field. Appgeometrical model of combinatorial eld established in Sucharacterize these combinatorial elds M (t) of gravitational

Yang-Mills eld M 1, M 2, · · ·, M n by tensor equations followi

Combinatorial Gravitational Field:

For a gravitational eld, we have known its Lagrange de

L = R −2κL F ,

where R is the Ricci scalar curvature, κ = −8πG and L F thall other elds. We have shown in Theorem 7 .2.1 that by thisEuler-Lagrange equations of gravitational eld are tensor equ

Rμν −12

gμν R = κE μν .

Now for a combinatorial eld M (t) of gravitational elthe combinatorial geometrical model established in Section 8 .its Lagrange density L

GL [M ] to be

L = R −2κL F ,

where

R = g(μν )( κλ )R (μν )( κλ ) , R (μν )( κλ ) = Rσς (μν )( σ

Then by applying the Euler-Lagrange equation we get t




since R(μν )( κλ ) |C i = Rμν , R|C i = R, g(μν )( κλ ) |C i = gμν and E (μν )( κλ )

Certainly, the equations (8 −10) can be also established likewWe will nd special solutions of (8 −10) in Section 8.3.

Combinatorial Yang-Mills Fields.

We have known the Lagrange density of a Yang-Mills eld is

L =12

tr( F μν F μν ) = −14

F iμν F iμν

with equations

D μF aμν = 0 and Dκ F aμν + DμF aνκ + Dν F aκμ = 0

For a combinatorial eld M (t) of gauge elds M 1, M 2, · · ·,Lagrange density L

GL [M ] to be

L GL [M ] =

12

tr( F (μν )( κλ ) F (μν )(κλ )) = −14

F ι(μν )( κλ ) F ι (μν

Then applying the Euler-Lagrange equation (8 −4), we can estabof combinatorial Yang-Mills eld as follows.

D μν F (μν )( στ ) = 0 and Dκλ F (μν )( στ ) + Dμν F (στ )( κλ ) + Dστ F

As a special case of the equations of combinatorial Yang-Millquently get the equations of combinatorial Maxwell eld following

∂ μν F (μν )( στ ) = 0 and ∂ κλ F (μν )( στ ) + ∂ μν F (στ )( κλ ) + ∂ στ F (κ

It should be noted that D μν |M i = D μ , F (μν )( στ ) |M i = F μν ,




combinatorial eld M of gravitational elds M i , 1 ≤ i ≤M j , 1 ≤ j ≤s with GL [M ] = GLS + GL

D , where GLS , GL

D aresubgraphs of gravitational elds or Yang-Mills elds in GL

Lagrange density L GL [M ] to be a linear combination

L GL [M ] = R −2κL F + 1

2tr( F μν F μν ) +

(M i ,M j )∈E (GLS ,G L

D

with

φM =k

i=1ciφM i +

s

j =1c j ψM j ,

where κ, bij , ci , c j are constants for 1 ≤ i ≤ k, 1 ≤ jequation by the Euler-Lagrange equation, or directly by the following:

R (μν )( στ ) −12g(μν )( στ )R + κE (μν )( στ ) + Dμν F

(μν )( στ )

−(M i ,M j )∈E (GL

S ,G LD )

bijψM

ci

For simplicity, let ci = c j = 1 for 1 ≤ i ≤k, 1 ≤ j ≤s. Aprinciple on gravitational elds M i , 1

≤i

≤k, we nd that

R(μν )( στ ) −12

g(μν )( στ ) R + κE (μν )( στ ) −(M i ,M j )∈E (GL

S ,G




Dμν F (μν )( στ ) −(M i ,M j )∈E (GL

S ,G LD )

bij ψM j = 0 .

Particularly, if we apply the projective principle on a Yang-Mills eld M j 0 for an integer j 0, 1 ≤ j0 ≤s, we get that

Dμ F μν −(M i ,M j 0 )∈E (GLS ,G L

D )bij ψM j 0 = 0,

∂ μ F μν −(M i ,M j 0 )∈E (GLS ,G L

D )bij ψM j 0 = 0

for Dμν |M j 0 = Dμ and Dμν |M j 0 = ∂ μ if M j 0 is a Maxwell eldcase of bij = 0, i.e., there are no actions between gravitational eldselds, we get the system of Einstein’s and Yang-Mills equations

R (μν )( στ ) −12

g(μν )( στ ) R = κE (μν )( στ ) , Dμν F (μν )( στ

§8.3 COMBINATORIAL GRAVITATIONAL FIELDS

For given integers 0 < n 1 < n 2 < · · ·< n m , m ≥ 1, a comtional eld M (t) is a combinatorial Riemannian manifold ( M,M (n1, n2,

· · ·, nm ) determined by tensor equations

R(μν )( στ ) −12

g(μν )( στ ) R = −8πG E (μν )( στ ) .

We nd their solutions under additional conditions in this section

Sec.8.3 Combinatorial Gravitational Fields



with xis

= x js

for 1 ≤ i, j ≤ s( p), 1 ≤ s ≤ s( p). A combinaby

ds2 = g(μν )( κλ )dxμν dxκλ ,

where g(μν )( κλ ) is the Riemannian metric in ( M,g, D). Genorthogonal basis {e11 , · · ·, e1n 1 , · · ·, es( p)n s ( p ) }for ϕ p[U ], p∈δ(κλ )

(μν ) . Then the formula (8 −11) turns to

ds2 = g(μν )( μν ) (dxμν )2

=s ( p)

μ=1 s( p)

ν =1

g(μν )( μν ) (dxμν )2 +s ( p)

μ=1 s( p)+1

ν =1

g(μν )( μν

=1

s2( p) s( p)

ν =1

(s ( p)

μ=1

g(μν )( μν ) )dxν +s( p)

μ=1 s ( p)+1

ν =1

g(μ

Then we therefore nd an important relation of combinatoriits projections following.

Theorem 8.3.1 Let μ ds2 be the metric of φ− 1 p (B n μ ( p) ) for

Then

ds2 = 1ds2 + 2ds2 + · · ·+ s ( p)ds2.

Proof Applying the projective principle, we immediatel

μds2 = ds2|φ− 1p (B n μ ( p ) ) , 1 ≤μ ≤s( p




8.3.2 Combinatorial Schwarzschild Metric. Let M be a gWe have known its Schwarzschild metric, i.e., a spherically symmEinstein’s gravitational equations in vacuum is

ds2 = 1 −r s

rdt2 −

dr 2

1

−r s

r−r 2dθ2 −r 2 sin2 θdφ2

in last chapter, where r s = 2 Gm/c 2. Now we generalize it to comtional elds to nd the solutions of equations

R (μν )( στ ) −12

g(μν )( στ )R = −8πG E (μν )( στ )

in vacuum, i.e., E (μν )( στ ) = 0. By the Action Principle of Fields inthe underlying graph of combinatorial eld consisting of m graa complete graph K m . For such a objective, we only considercombinatorial Euclidean spaces M = m

i=1 R n i , i.e., for any point

[ϕ p] =

⎡⎢⎢⎢⎢⎣

x11

· · ·x1

m x1(

m )+1)

· · ·x1n 1

· · ·x21 · · · x2 m x2( m +1) · · · x2n 2 · · ·· · · · · · · · · · · · · · · · · ·xm 1 · · · xm m xm ( m +1) · · · · · · · · ·xm

with m = dim(m

i=1R n i ) a constant for ∀ p∈

m

i=1R n i and xil = x l

m f

l ≤m.Let M (t) be a combinatorial eld of gravitational elds M 1

masses m1, m2, · · ·, mm respectively. For usually undergoing, we of n = 4 for 1 ≤μ ≤m since line elements have been found con




where μds2

= ds2μ by the projective principle on combinato

1 ≤m ≤4. We therefore get combinatorial metrics dependen

Case 1. m = 1 , i.e., tμ = t for 1 ≤μ ≤m.

In this case, the combinatorial metric ds is

ds2 =m

μ=1

1 −2Gmμ

c2r μdt2 −

m

μ=1

(1 −2Gm μ

c2r μ)− 1dr 2

μ −m

μ=1

Case 2. m = 2 , i.e., tμ = t and r μ = r , or tμ = t and θμ = for 1

≤μ

≤m.

We consider the following subcases.

Subcase 2.1. tμ = t, r μ = r .

In this subcase, the combinatorial metric is

ds2 =m

μ=1

1 −2Gm μ

c2rdt2 −(

m

μ=1

1 −2Gm μ

c2r− 1

)dr 2 −m

μ

which can only happens if these m elds are at a same pointlarly, if mμ = M for 1 ≤μ ≤m, the masses of M 1, M 2, · · ·,r μg = 2GM is a constant, which enables us knowing that

ds2 = 1 −2GM

c2rmdt 2 − 1 −

2GM c2r

− 1

mdr 2 −m

1

r




Subcase 2.3. tμ = t, φμ = φ.In this subcase, the combinatorial metric is

ds2 =m

μ=1

1 −2Gm μ

c2r μdt2 −(

m

μ=1

1 −2Gm μ

c2r μ

− 1

)dr 2μ −

m

μ=1

r 2μ (

Case 3. m = 3 , i.e., tμ = t, r μ = r and θμ = θ, or tμ = t, r μ =or tμ = t, θμ = θ and φμ = φ for 1 ≤μ ≤m.

We consider three subcases following.

Subcase 3.1. tμ = t, r μ = r and θμ = θ.


ds2 =m

μ=1

1 −2Gm μ

c2rdt2 −

m

μ=1

1 −2Gm μ

c2r

− 1

dr 2 −mr 2dθ2

Subcase 3.2. tμ = t, r μ = r and φμ = φ.


ds2 =m

μ=1

1 −2Gm μ

c2rdt2 −

m

μ=1

1 −2Gm μ

c2r

− 1

dr 2 −r 2m

μ=1

(d

There subcases 3 .1 and 3.2 can be only happen if the centersare at a same point O in a space.




ds2 =m

μ=1

1 −2Gm μ

c2rdt2 −

m

μ=1

1 −2Gm μ

c2r

− 1

dr 2 −m

Particularly, if mμ = M for 1 ≤μ ≤m, we get that

ds2 = 1 −2GM c2r

mdt 2 − 1 −2GM

c2r

− 1

mdr 2 −mr

Dene a coordinate transformation ( t,r,θ,φ ) →( s t, sr,Then the previous formula turns to

ds2 = 1 −2GM

c2rds t2 −

dsr 2

1 − 2GM c2 r

− s r 2(ds θ2 +

in this new coordinate system ( s t, s r, s θ, s φ), whose geothat of the gravitational eld.

8.3.3 Combinatorial Reissner-Nordstr¨ om Metric. Tric is a spherically symmetric solution of the Einstein’s gravconditions E (μν )( στ ) = 0. In some special cases, we can alsothe case E (μν )( στ ) = 0. The Reissner-Nordstr¨ om metric is su

E (μν )( στ ) =1

4π

1

4gμν F αβ F αβ

−F μα F αν

in the Maxwell eld with total mass m and total charge e, given in Subsection 7.3.4. Its metrics takes the following for

⎡




Obviously, if e = 0, i.e., there are no charges in the gravitationequations (8 −13) turns to the Schwarzschild metric (8 −12).

Now let M (t) be a combinatorial eld of charged gravitational M m with masses m1, m2, · · ·, mm and charges e1, e2, · · ·, em , respethe case of Schwarzschild metric, we consider the case of nμ = 4 festablish m spherical coordinate subframe ( tμ ; r μ , θμ , φμ ) with itscenter of such a mass space. Then we know its a spherically symm(8 −13) to be

ds2μ = 1

−

r μs

r μ

+r 2

μe

r2μ

dt2μ

−1

−

r μs

r μ

+r 2

μe

r2μ

− 1

dr 2μ

−r 2

μ (dθ2μ

Likewise the case of Schwarzschild metric, we consider comcharged gravitational elds dependent on the intersection dimensio

Case 1. m = 1 , i.e., tμ = t for 1 ≤μ ≤m.

In this case, by applying Theorem 8 .3.1 we get the combinat

ds2 =m

μ=1

1 −r μs

r μ+

r 2μe

r 2μ

dt2−m

μ=1

1 −r μs

r μ+

r 2μe

r 2μ

− 1

dr 2μ−

m

μ=1

r

Case 2. m = 2 , i.e., tμ = t and r μ = r , or tμ = t and θμ = θ, or for 1 ≤μ ≤m.

Consider the following three subcases.




ds2 = 1 −2GM

c2r+

4πGe 4

c4r 2 mdt 2 −mdr 2

1 − 2GM c2 r + 4πGe 4

c4 r 2 −m

μ=

Subcase 2.2. tμ = t, θμ = θ.

In this subcase, by applying Theorem 8 .3.1 we know metric is

ds2 =m

μ=1

1 −r μs

r μ+

r 2μe

r 2μ

dt2−m

μ=1

1 −r μs

r μ+

r 2μe

r 2μ

− 1

dr 2μ−

Subcase 2.3. tμ = t, φμ = φ.

In this subcase, we know that the combinatorial metric i

ds2

=

m

μ=1 1 −r μs

r μ +r 2

μe

r 2μ dt

2

−m

μ=1 1 −r μs

r μ +r 2

μe

r 2μ

− 1

dr2μ−

Case 3. m = 3 , i.e., tμ = t, r μ = r and θμ = θ, or tμ = t,or tμ = t, θμ = θ and φμ = φ for 1 ≤μ ≤m.

We consider three subcases following.

Subcase 3.1. tμ = t, r μ = r and θμ = θ.

In this subcase, by applying Theorem 8 .3.1 we obtain




Subcase 3.2. tμ = t, r μ = r and φμ = φ.In this subcase, the combinatorial metric is

ds2 =m

μ=1

1 −r μs

r+

r 2μe

r 2 dt2−m

μ=1

1 −r μs

r+

r 2μe

r 2

− 1

dr 2−m

μ=1

r

Particularly, if mμ = M and eμ = e for 1 ≤μ ≤m, then we get th

ds2 = 1 −2GM

c2r+

4πGe 4

c4r 2 mdt 2 −mdr 2

1 − 2GM c2 r + 4πGe 4

c4 r 2 −m

μ=1

r 2(

Subcase 3.3. tμ = t, θμ = θ and φμ = φ.


ds2

=

m

μ=1 1 −r μs

r μ +r 2

μe

r 2μ dt2

−m

μ=1 1 −r μs

r μ +r 2

μe

r 2μ

− 1

dr2μ−

m

μ=1

Case 4. m = 4 , i.e., tμ = t, r μ = r, θ μ = θ and φμ = φ for 1 ≤μ


ds2 =m

μ=1

1 −r μs

r+

r 2μe

r 2 dt2

m 2 − 1




ds2 = 1 −2GM c2r

+4πGe 4

c4r 2 ds t2 −ds r 2

1 − 2GM c2 r + 4πGe 4

c4 r 2 − s r

in this new coordinate system ( s t, s r, s θ, s φ), whose geothat of a charged gravitational eld.

8.3.4 Multi-Time System. Let M (t) be a combinatorial M 1, M 2, · · ·, M m on reference frames (t1, r 1, θ1, φ1), · · ·, (tm , rThese combinatorial elds discussed in last two subsectionfor 1 ≤ μ ≤ m, i.e., we can establish one time variablecombinatorial eld. But if we can not determine all the behavthe WORLD implied in the weak anthropic principle , for exof micro-particles, we can not nd such a time variable t for a multi-time system for describing the WORLD.

A multi-time system is such a combinatorial eld MM 1, M 2, · · ·, M m on reference frames (t1, r 1, θ1, φ1), · · ·, (tm

are always exist two integers κ, λ, 1 ≤κ = λ ≤m such thatical meaning of multi-time systems is nothing but a refection principle. So it is worth to characterize such systems.

For this objective, a more interesting case appears inθ, φμ = φ, i.e., beings live in the same dimensional 3 sp

notions on the time. Applying Theorem 8 .3.1, we knowReissner-Nordstr¨om metrics in this case following.

Schwarzschild Multi-Time System. In this case, the co




ds22 = 1 −

2Gm 2

c2rdt2

2 − 1 −2Gm 2

c2r

− 1

dr 2 −r 2(dθ2 +

· · · · · · · · · · · · · · · · · · · · · · · · · ·,

ds2m = 1 −

2Gm m

c2rdt2

m − 1 −2Gm m

c2r

− 1

dr 2 −r 2(dθ2 +

Particularly, if mμ = M for 1 ≤μ ≤m, we then get that

ds2 = 1 −2GM c2r

m

μ=1dt2μ − 1 −2GM c2r

− 1

mdr 2 −mr 2(dθ

Its projection on the gravitational eld M μ is

ds2μ = 1

−2GM

c2

rdt2

μ

−1

−2GM

c2

r

− 1

dr 2

−r 2(dθ2 +

i.e., the Schwarzschild metric on M μ , 1 ≤μ ≤m.

Reissner-Nordstr¨ om Multi-Time System. In this case, metric is

ds2 =m

μ=1

1 −2Gm μ

c2r+ 4πGe 4μ

c4r 2 dt2μ

m

12Gm μ +

4πGe 4μ

− 1

d 2 2(dθ2 +




· · · · · · · · · · · · · · · · · · · · · · · ·,

ds 2m = 1 −

2Gm m

c2r+

4πGe 4m

c4r 2 dt 2m − 1 −

2Gm 2

c2r+

4πGe 42

c4r 2

− 1

Furthermore, if mμ = M and eμ = e for 1 ≤μ ≤m, we obta

ds2 = 1 −2GM

c2r+

4πGe 4

c4r 2

m

μ=1

dt2

− 1 −2GM

c2r+ 4πGe 4

c4r 2

− 1mdr 2 −mr 2(dθ

Its projection on the charged gravitational eld M μ is

ds2μ = 1 −

2GM c2r +

4πGe 4

c4r 2 dt2μ− 1 −

2GM c2r +

4πGe 4

c4r 2

− 1

i.e., the Reissner-Nordstr¨om metric on M μ , 1 ≤μ ≤m.As a by-product, these calculations and formulas mean t

time notion different from that of human beings will recognizture of our universe if these beings are intellectual enough to

8.3.5 Physical Condition. A simple calculation showsthe homogenous combinatorial Euclidean space M (t) in Sub




references [Rei1], [Yaf1] and [Bel1] in details. In these geometricathose of point, line, ray, block, body, · · ·, etc., we can only see thon our spherical surface, i.e., surface blocks.

Secondly, what is the geometry of transferring information information includes information known or not known by humangeometry of transferring information consists of all possible transfother words, a combinatorial geometry of dimensional ≥1. Theremation transferring can be seen by our eyes. But some of them csix organs with the helps of apparatus if needed. For example, telectromagnetism can be only detected by apparatus.

These geometrical notions enable us to explain the physical co

binatorial metrics, for example, the Schwarzschild or Reissner-Nor

Case 1. m = 4.

In this case, by the formula (8 −14) we get that dim M (t) M 1, M 2, · · ·, M m are in R 4, which is the most enjoyed case by h

have gotten the Schwarzschild metric

ds2 =m

μ=1

1 −2Gm μ

c2rdt2 −

m

μ=1

1 −2Gm μ

c2r

− 1

dr 2 −mr 2(d

for combinatorial gravitational elds or the Reissner-Nordstr¨ om m

ds2 =m

1 −r μs +

r 2μe2 dt2 −

dr 2

m 2 −mr 2(d




But unfortunately, the existence of Theory of Everythinweak anthropic principle, and more and more natural phenomWORLD is a multiple one. Whence, this case maybe wrong.

Case 2. m ≤3.

If the WORLD is so, then dim M (t)

≥5. In this ca

binatorial Schwarzschild metrics and combinatorial Reissner-Subsection 8.2.2−8.2.3, for example, if tμ = t, r μ = r and φμ

Schwarzschild metric is

ds2 =m

μ=1

1

−r μs

rdt2

−

m

μ=1

dr 2

1 − r μsr −

m

μ=1

r 2(dθ

and the combinatorial Reissner-Nordstr¨ om metric is

ds2 =m

μ=1

1 −r μs

r+

r 2μe

r 2 dt2 −m

μ=1

dr 2

1 −r μs

r +r 2

μe

r 2

−m

μ=

Particularly, if mμ = M and eμ = e for 1 ≤μ ≤m, then we

ds2 = 1 −2GM

c2rmdt 2 −

mdr 2

1 − 2GM c2 r

−m

μ=1

r 2(dθ

for combinatorial gravitational eld and

4 2




beings. In this case, we are difficult to determine the exact behaviopartial information of the WORLD, which means that each law determined by human beings is an approximate result and hold wi

Furthermore, if m ≤ 3 holds, since there are innite undgraphs, i.e., there are innite combinations of existent elds, weultimate theory for the WORLD, i.e., there are no a Theory of WORLD. This means the science is approximate and only a realstraint on conditions, which also implies that the discover of sciforever.

§8.4 COMBINATORIAL GAUGE FIELDS

A combinatorial gauge eld M (t) is a combinatorial eld of gau

· · ·, M m underlying a combinatorial structure G with local or gunder a nite-dimensional Lie multi-group action on its gauge basis

point in space and time, which leaves invariant of physical laws,Lagrange density L of M (t). We mainly consider the followinsection.

Problem 8.4.1 For gauge elds M 1, M 2, · · ·, M m with respective L M 1 , L M 2 , · · ·, L M m and action by Lie groups H ◦ 1 , H ◦ 2 , · · ·, H

on L GL [M ](t) , the Lie multi-group H and GL [M (t)] such that the cM (t) consisting of M 1, M 2, · · ·, M m is a combinatorial gauge led

8 4 1 Gauge Multi-Basis For any integer i 1 ≤ i ≤m let

Sec.8.4 Combinatorial Gauge Fields



for integers 1 ≤ i ≤m such that

L M (

m

i=1

BM i )τ M = L M (

m

i=1

BM i ).

By Theorem 3.1.2 the Gluing Lemma, we know that ii.e., τ i |B M i ∩B M j

= τ j |B M i ∩B M jfor all integers 1 ≤ i, j ≤m, the

continuous τ M : m

i=1BM i →m

i=1BM i with τ M |M i = τ i for all in

Notice that τ i |B M i ∩B M j= τ j |B M i ∩B M j

hold only if (BM i

for any integer 1 ≤ j ≤m. This is hold in condition. For exidentity mapping, i.e., τ i = 1 B M i

, 1 ≤ i ≤m, then it is obviouBM i

∩BM j , and furthermore, τ i

|B M i ∩B M j

= τ j

|B M i ∩B M j

for in

Now we dene a characteristic mapping χ M i on {BM i ;

χ M i (X ) =1, if X = BM i ,0, otherwise .

Then

τ M =m

i=1

χ M i τ i .

In this case, the Lagrange density

L M =

m

i=1

χ M i L M i

on M holds with L M |M i = L M i for each integer 1 ≤ i

M i = M, 1 ≤ i ≤m, then




(χ M 1 + χ M 2 + · · ·+ χ M n )(BM i ) = 1

for integers 1 ≤ i ≤m, but it maybe not a constant onm

i=1BM i for

of elds M i , 1 ≤ i ≤m in space.Let the motion equation of gauge elds M i be F i (L M i ) =

Applying Theorem 7 .1.6, we then know the eld equation of comof M 1, M 2, · · ·, M m to be

χ M 1 F 1(L M 1 ) + χ M 2 F 2(L M 2 ) + · · ·+ χ M m F m (L M m

for the linearity of differential operation ∂/∂φ . For example, let

torial eld consisting of just two gauge eld, a scalar eld M 1 andThen the eld equation of M is as follows:

χ M 1 (∂ 2 + m2)ψM 1 + χ M 2 (iγ μ∂ μ −m)ψM 2 = 0 .

8.4.2 Combinatorial Gauge Basis. Let M be a combinatoelds M 1, M 2, · · ·, M m . The multi-basis

m

i=1BM i is a combinatorial

any automorphism g∈Aut GL [M ],

L M (

m

i=1

BM i )τ M ◦ g = L M (

m

i=1

BM i ),

where τ M ◦g means τ M composting with an automorphism g, a b

multi-basism

i=1BM i . Now if Ω1, Ω2, · · ·, Ωs are these orbits of eld

d th ti f A t GL [M ] th th t b th t




L M =

s

α =1|(

M αi ∈Ωα

χ M αi )L M i

on M holds with L M |M i = L M i for each integer 1 ≤ i ≤m.

We discussed two interesting cases following.

Case 1. GL [M ] is transitive.

Because GL [M ] is transitive, there are only one orbit ΩWhence, M i = M for integers 1 ≤ i ≤m, i.e., the combinatorof one gauge eld M underlying a transitive graph GL [M ].

In this case, we easily know that

m

i=1

BM i = BM ,

τ M = ( χ M 1 + χ M 2 + · · ·+ χ M m )τ M

and

L M = ( χ M 1 + χ M 2 + · · ·+ χ M m )L M ,

which is the same as the case of gauge multi-basis with a com

Case 2. GL [M ] is non-symmetric.

Since GL [M ] is non-symmetric, i.e., Aut GL [M ] is trivial,M m are distinct two by two. Whence, the combinatorial of gauge elds M 1, M 2, · · ·, M m underlying a non-symmetrτi |B B = τj |B B for all integers 1 ≤ i j ≤m




of gauge elds M 1, M 2, · · ·, M m is a combinatorial gauge eld ∀M α1 , M α2 ∈Ωα , where Ωα , 1 ≤ α ≤ s are orbits of M 1, M 2, ·action of Aut GL [M ]. In this case, each gauge transformation caby τ ◦g, where τ is a gauge transformation on a gauge eld Mg∈Aut GL [M ] and

τ M =s

α =1

(M αi ∈Ωα

χ M αi )τ i , L M =

s

α =1|(

M αi ∈Ωα

χ M αi )

All of these are dependent on the characteristic mapping χ M i ,difficult for use. Then

whether can we construct the gauge transformation τ M andL

M independent on χ M i , 1 ≤ i ≤m?

Certainly, the answer is YES! We can really construct locally comelds by applying embedded graphs on surfaces as follows.

Let ς : GL [M ] →S be an embedding of the graph GL [M ] oa compact 2-manifold without boundary with a face set

F =

{F 1

By assumption, if ( M i1 , M i2 )∈E (GL [M ]), then M i1 ∩M i2 is also the action of τ i1 |M i 1 ∩M i 2

= τ i2 |M i 1 ∩M i 2. Whence, we can always

density L M i 1 ∩M i 2.

Now relabel vertices and edges of GL [M ] by

M Li = L M i , (M i , M j )L = L M i 1 ∩M i 2 for 1 ≤ i, j ≤with (M j , M i)L = −(M i , M j )L , and if F i = M i1 M i2 · · ·M is , then la


L L



ι :L

M i ∩M j →L

M i ∩M j + φ j (t) −φi(t),ι : LM i (t) →LM i (t) + φi(t),

where φi(t) is a function on eld M i , 1 ≤ i ≤m. Calculation

ι :˙L M i ∩M j +

L M i −

L M j

→ ˙L M i ∩M j + φ j (t) −φi(t) + L M i + φi(t) −L

= ˙L M i ∩M j + L M i −L M j

and

ι : F Li = L M i 1 ∩M i 2+ L M i 2 ∩M i 3

+ · · ·+ L M i s ∩M i 1

→L M i 1 ∩M i 2+ φi2 (t) −φi1 (t) + · · ·+ L M i s ∩M i 1

+

= L M i 1 ∩M i 2+ L M i 2 ∩M i 3

+ · · ·+ L M i s ∩M i 1= F Li

Therefore, L ιM = L

M , i.e., ι is a gauge transformation. Thused to describe physical objectives. For example, let GL [partially show in Fig.8 .4.1 following,

i j




8.4.4 Geometry on Combinatorial Gauge Field. We havometrical model of combinatorial eld in Subsection 8 .1.3. Comwith combinatorially principal ber bundles discussed in Section 6lish a geometrical model of combinatorial gauge eld, which also ewhat is the gauge basis of a combinatorial gauge eld.

Likewise the geometrical model of gauge eld, let P α (M,natorially principal bre bundle over a differentiably combinatoconsisting of M i , 1 ≤ i ≤ l, (GL [M ], α ) a voltage graph with ment α : GL [M ] →G over a nite group G , which naturally indπ : GL [P ] →GL [M ] and P M i (M i , H ◦ i ), 1 ≤ i ≤ l a family of prinover manifolds M 1, M 2, · · ·, M l . By Construction 6 .5.1, P α (M, L

by for∀M ∈V (GL [M ]), place P M on each lifting vertex M L α inof GL α [M ] if π(P M ) = M . Consequently, we know that

P =M ∈V (GL [M ])

P M , L G =M ∈V (GL [M ])

H M

and a projection Π = πΠM π− 1 for∀M ∈V (GL [M ]). By denitionprincipal ber bundle P α (M, L G ) is AutGL α [M ]×L G -invariant.a combinatorial gauge eld under the action of Aut GL α [M ] ×L

gauge and gauge transformations rst.For a combinatorial principal ber bundle P α (M, L G ), we

trivialization LT is such a diffeomorphism T x

: Π− 1

(U x ) →U x ×with

T x |Π− 1i (U x ) = T xi : Π− 1

i (U x ) →U x ×H ◦ i ; x →T xi (x) = (Π


Notice that an automorphism of P can not ensure the



Notice that an automorphism of P can not ensure the density L

M in general. A gauge transformation of P α (M, L

phism ω : P →P with ω =identity transformation on M , i p∈P . Whence, L

M is invariant under the action of ω, i.e.,

L M

(l

i=1

BM i )ω = L M

(l

i=1

BM i ).

As we have discussed in Subsections 6.5.3 −6.5.4, therecome from two sources. One is the gauge transformations τ M1 ≤ i ≤ l. Another is the symmetries of the lifting graph GL α

inner symmetries to the outer in a combinatorial eld. When

principal ber bundle enables us to nd more gauge elds foNow let

1ω be the local connection 1-form,

2Ω= d

1ω the

local connection on P α (M, L G ) and Λ : M →P , Π ◦Λ section of P α (M, L G ). Similar to that of gauge elds, we co

A = Λ∗1ω=

μν

Aμν dxμν ,

F = Λ∗2Ω= F (μν )( κλ ) dxμν

∧dxκλ , d F

which are called the combinatorial gauge potential and combrespectively. Let γ : M →R and Λ : M →P , Λ (x) = eiγ

then we have1

ω (X ) = g− 1 1ω (X )g + g− 1dg, g∈

L


1 2 1



Now if we chooseω and Ω= d ω be the global connection 1-fo2-form of a global connection on P α (M, L G ), respectively, we can equations (8 −15) by applying properties of global connections oprincipal ber bundle P α (M, L G ) established in Section 6 .5, andto determine the behaviors of combinatorial gauge elds.

8.4.5 Higgs Mechanism on Combinatorial Gauge Field. Luum state in a combinatorial gauge eld M consisting of gauge eldwith the Lagrangian L

M = L 1 + V M (ΦM ), where V M (ΦM ) stands potential in M , Aut GL [M ]×L G a gauge multi-group and g →ϕ(gof AutGL [M ]×L G . Dene

ΦM 0 = ϕ(Aut GL [M ]×L G )ΦM 0 = {ϕ(g)ΦM 0 |g∈Aut GL [M ]

and (Aut GL [M ] ×L G )0 = {g∈Aut GL [M ] ×L G|ϕ(g)ΦM 0 = Φcalled a homogenous space of AutGL [M ]×L G , that is,

M 0 = Aut GL [M ]×L G / (Aut GL [M ]×L G

= {ϕ(g)ΦM 0 |g∈Aut GL [M ]×L G}.

Similarly, a gauge symmetry in Aut GL [M ]×L G associated wigauge eld is said to be spontaneously broken if and only if t

manifold M 0 dened in (8 −17) gotten by a vacuum state Φ M 0 dFurthermore, if we let V M (ΦM 0 ) = 0 and V M (ϕ(g)ΦM ) = V (ΦM )be V M (ϕ(g)ΦM 0 ) = 0. Therefore, we can rewrite M 0 = {ΦM |V M (Φ




ΦM 0 =m

i=1

χ M i ΦM 0i .

Then we get that

V M (ΦM 0 ) = V M (

m

k=1 χ M k ΦM 0k )

=m

k=1

χ M k V M k (m

i=1

χ M i ΦM

=m

k=1

χ M k V M k (ΦM 0k ) = 0 .

Conversely, if V M (ΦM 0 ) = 0, then V M (ΦM 0 )|M i = 0, ifor integers 1 ≤ i ≤ m. Let M 0i = M 0|M i . Then ΦM 0i = V M i (ΦM 0i ) = 0, i.e.,

ΦM 0 ) =m

i=1

χ M i ΦM 0i .

Summing up all discussion in the above, we get the nex

Theorem 8.4.1 Let M be consisting of gauge elds M 1, Mgrangian L

M = L 1 + V M (ΦM ). If ΦM 0 is its vacuum statΦM 0i , 1

≤i

≤m, Then M 0 is a combinatorial eld consisting

Particularly, if M i = M for integers 1 ≤ i ≤m, then we

V (Φ ) = 0 (m

χ )V (Φ ) =


§8.5 APPLICATIONS



§8.5 APPLICATIONS

The multi-laterality of WORLD alludes multi-lateralities of things also more applicable aspects of combinatorial elds. In fact, as we the behavior of a family of things with interactions, the best modenothing but a combinatorial eld.

8.5.1 Many-Body Mechanics. The many-body mechanicsprovides the framework for understanding the collective behavior oof interacting particles, such as those of solar system, milky whave known a physical laws that govern the motion of an individbe simple or not, but the behavior of collection particles can be exFor being short of mathematical means on many-bodies over a lonone know few laws of many-body systems, even for two-body sys

Let M be a many-body system consisting of bodies M 1, M 2ample, let M be the solar system, then M 1 =Sun, M 2 =MercM 4 =Earth, M 5 =Mars, M 6 =Jupiter, Saturn, M 7 =Uranus an

such as those shown in Fig. 8 .5.1.

Sun

Mercury

VenusEarth

Jupiter

SaturnUranus Nep

Sec.8.5 Applications

development in R 3 on the time t.



pNotice that the solar system is not a conservation sys

system. As we turn these actions between planets to internaare still external actions coming from other planets not in soonly nd an approximate model by combinatorial eld. Mothe universe beyond the solar system, for example, a combiway, then more accurate result on the behavior of solar syste

8.5.2 Cosmology. Modern cosmology was established upoativity, which claims that our universe was brought about a Bipoint, the time began. But there are no an argument explainonce. It seems more reasonable that exploded many times if oto happen for the WORLD. Then the universe is not lonelywith other universes. If so, a right model of the WORLD shoone U consisting of universes U 1, U 2, · · ·, U n for some integerunverse brought about by the ith Big Bang, a manifold in m

Applying the sheaf structure of space in algebraic geometr

for the universe was given in references [Mao3] and [Mao10]with combinatorial elds, we present a combinatorial model of

A combinatorial universe is constructed by a triple (Ω ,

Ω =i≥ 0

Ωi , Δ =i≥ 0

Oi

and T = {t i ; i ≥0}are respectively called the universes, theset with the following conditions hold.


(i) for (Ωi1, Oi), (Ωi2, Oi)(Ωi3, Oi) (S ), if Ωi1 Ωi2 Ωi3



( )

∀

( ) ( )( )

∈

( )

⊃ ⊃

ρΩi 1 ,Ωi 3 = ρΩi 1 ,Ωi 2 ◦ρΩi 2 ,Ωi 3 ,

where ¡ ◦¢ denotes the composition operation on homomorphisms.

(ii ) for∀g, h∈Ωi , if for any integer i, ρΩ,Ωi (g) = ρΩ,Ωi (h), t

(iii ) for∀i, if there is an f i∈Ωi with

ρΩi ,Ωi Ωj (f i) = ρΩj ,Ωi Ωj (f j )

for integers i,j, Ωi Ω j = ∅, then there exists an f ∈Ω such thaany integer i.

If we do not consider its combinatorial structure GL [M ], M (ta multi-space. Because the choice of GL [M ] and integer n is establish innite such combinatorial models for the universe. Thein front of us is to determine which is the proper one.

Certainly, the simplest case is |GL [M ]| = 1, overlooking

structure GL [M ]. For example, for dimensional 5 or 6 spaces, it haa dynamical theory in [Pap1]and [Pap2]. In this dynamics, we loin the Einstein’s equation of gravitational eld in 6-dimensional metric of the form

ds2 = −n2(t ,y,z )dt2 + a2(t ,y,z )d2

k

+ b2(t ,y,z )dy2 + d2

2




d2

k

=dr 2

1 −kr 2 + r 2dΩ2(2) + (1 −kr 2)d

h(z) = k +z2

l2 −M z3

and the energy-momentum tensor on the brane is

T μν = hνα T αμ −14

T hμν

with T αμ = diag (−ρ,p,p,p, ˆ p). Then the equation of a 4-dimenin a 6-spacetime is

2¨RR + 3(

˙RR )2 = −3

κ4

(6)64 ρ2 −

κ4

(6)8 ρp −3 κR2

by applying the Darmois-Israel conditions for a moving branewhere K μν is the extrinsic curvature tensor. Similarly, for ththe equations of motion of the brane are

d2dR −dR

1 + d2R2 − 1 + d2R2

n(dnR +

∂ z nd −(d∂ zn −n∂ z d)R2)

∂ zaad 1 + d2R2 = −

κ4(6)

8(ρ + p −ˆ p),

∂ z bbd 1 + d2R2 = −

κ4(6)

8(ρ −3( p−ˆ p))

Problem 8 5 1 Establish dynamics of combinatorial universe


n

i



φM = i=1 ciφM i ;

L GL [M ] =

n

i=1ψM i (iγ μ i ∂ μ i −m i)ψM i +

(M i ,M j )∈E (GL [M ])bij ψM

where bij , mi , ci , C are constants for integers 1 ≤ i, j ≤n and w

n

i=1

1ci

(iγ μ i ∂ μ −m i)ψM i −(M i ,M j )∈E (GL [M ])bij

ψM j

ci+

ψc

the equation of eld established in Subsection 8 .2.3.Another application of combinatorial eld to physical structure

a model for atoms and molecules. As we said, the combinatorial

a physical model for many-body systems, which can naturally be umany-body system, such as those of atoms, molecules and other su

8.5.4 Economical Field. An economical eld is an organized syarrangement of parts. Let P 1(t, x ), P 2(t, x ), · · ·, P s (t, x ), s ≥1 be pfactors x in an economical eld E S . Certainly, some of P 1(t, x ), P

may be completely or partially conned by others. If we view eacbe a eld, or a smooth manifold in mathematics, then E S is a coconsisting of elds P 1(t, x ), P 2(t, x ), · · ·, P s (t, x ). Therefore, we such as those of differential properties on combinatorial manifolds6 to grasp the behavior of an economical eld and then releaseforecasting for regional or global economy.

As a special case, a circulating economical eld is a combinconsisting of economical elds M 1(t), M 2(t), · · ·, M s(t) underlying C for an integer s 2 such as those shown in Fig 8 5 2


Today, nearly every regional economy is open with inte



velopment of economy, an urgent thing is to set up a conservabeing with nature in harmony, i.e., to make use of matter aand everlastingly £ to decrease the unfavorable effect that ecmake upon our natural environment as far as possible, whichcirculating eld for the global economy following.

¹

»

Ó

Utility resources

Green product Recyclic resour

Fig. 8.5.3

Whence, we can establish a combinatorial model consisting of

in our society. Therefore, we decide the economic growth by combinatorial differential geometry in Chapters 5 −6,the development of human being’s society harmoniously withwhich can be determined if all factors in this economical eld are known. That is a global economical science for our soresearch furthermore.

Besides applications of combinatorial elds to physicsare many other aspects for which combinatorial elds can be



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Index combinatorial concombinatorial · ·



A

action at a distance 362action of Lie multi-group 324action of multi-group 70action principle of eld 416adjoint representation 318algebraic multi-system 78algebraic subsystem 49algebraic system 46associative law 46automorphism of · · ·

multi-system 54principal ber bundle 332

system 49

B

basis 113Boolean algebra 5Boolean polynomials 8boundary homomorphism 105bracket operation 226

connection spcurvature opdifferential mDirac eld

embedded suequivalence Euclidean spfan-space 1ber bundle

eld 414eld strengthFinsler geomfree-eld 4gauge basis

gauge eld gauge potentGauss theorgravitational in-submanifomanifold 1

combinatorial · ·metric 439

478 Linfan Mao: Combinatorial Geometry with Applications t

system 43Yang-Mills eld 436

edge labeled graph 2Einstein gravitational e



commutative law 46complete graph 24connection on tensor 238conservation law 360

continuous mapping 90contractible space 188coordinate matrix 224co-tangent vector space 144, 230countable set 18

covering space 99curvature forms 249curvature forms

on principal ber bundle 342curvature tensor 247

D

d-connectedness 174deformation retract 188

differential · · ·k-form 122manifold 140mapping 118

electromotive force 3electromagnetic eld electromagnetic eld teelectrostatic eld 378

elliptic vertex 127, 13embedded graph 25Euclidean space 111Euclidean vertex 127Euler-Lagrange equatio

Euler-Poincare characteexact chain 201exact differential form exterior differential 1

F

nite combinatorial manite dimensional multfundamental d-group

fundamental group 95

fundamental theorem oLie group 318Lie multi-group

Index

Gauss formula 284global connection on

J

J bi i



principal ber bundle 336Gluing Lemma 90graph 19group 49

H

Hamiltonian eld 357Hamiltonian

on combinatorial eld 827

Hamiltonian principle 355Hasse diagram 12Higgs eld 404Higgs particle 404homology sequence 20homomorphic multi-system 54homomorphic system 49homotopic equivalence 183homotopic mapping 95 2

homotopy equivalence 195

hyperbolic vertex 127, 134, 138

I

Jacobian matrix

K

k-connected graph

Kruskal coordinat

L

labeled lifting 2Lagrange equation

Lagrange eld 3

Lagrangian oncombinatoria

Lebesgue LemmaLie algebra 312

Lie multi-group Lie multi-subgrou

lifting automorphvoltage label

lifting of a

voltage labellinear extension

linear mapping

480 Linfan Mao: Combinatorial Geometry with Applications t

magnetostatic eld 380map geometry · · ·

one-parameter multi-gropen sets 89



with boundary 132without boundary 128

mathematical system 41Maxwell equation 385

Maxwell led 394Mayer-Vietoris theorem 203metric space 92Minkowskian norm 145Minkowskian spacetime 364

multi-eld 58multi-group 58multi-ideal 60multilinear mappings 117multi-module 61

multi-operation system 53multi-poset 14multi-ring 58multi-set 9multi-subeld 58multi-subgroup 58multi-subring 58multi-time system 447

orbit 71ordered set 13

P

parking problem 155partially ordered set 1particle-antiparticle

transformation 3partition of unity 259Pauli matrix 396Poisson equation 363polyhedron 104power set 3principal ber bundle

principle of covarianceprinciple of equivalencprojective principle 4pseudo-Euclidean spacepseudo-manifold 138

pseudo-manifold geom

Q

Index

S

Schwarzschild metric 370

topological multi-topological multi-



Schwarzschild metric 370Schwarzschild radius 375Seifert and Van-Kampen 98simplicial homology group 106

simplex 103simply d-connected 195singular homology group 198, 207s-manifold 127Smarandache geometry 126Smarandache manifold 141Smarandache multi-space 43Smarandache system 42spanning subgraph 23spinor of eld 395spontaneous symmetry broken 403stabilizer 71standard n-dimensional poset 13standard p-simplex 123structural equation 249subgraph 22

subgroup 49subposet 13strong anthropic principle 421

topological multi-topological multi-topological space torsion-free 241

transitive multi-grtriangulation 109

U

unied eld 412

universal gravitati

V

valency sequence variation 351

vector eld 226vertex-edge labelevertex-induced grvertex labeled gravoltage labeled gr

W

weak anthropic pr

Linfan Mao is a researcher in the Chinese Academy of Math-

ematics and System Science , also a deputy general secretary

of the China Tendering & Bidding Association , in Beijing. He

got his Ph.D in Northern Jiaotong University in 2002 and n-

ished his postdoctoral report for Chinese Academy of Sciences



ished his postdoctoral report for Chinese Academy of Sciences

in 2005. His main interesting focus on Mathematical Combin-

atorics and Smarandache multi-spaces with applications to sciences, particularly,

the parallel universe.

ABSTRACT : This monograph is motivated with surveying mathematics and

physics by CC conjecture, i.e., a mathematical science can be reconstructed from or made by combinatorialization . Topics covered in this book include fundamental

of mathematical combinatorics, differential Smarandache n -manifolds, combinato-

rial or differentiable manifolds and submanifolds, Lie multi-groups, combinatorial

principal ber bundled, gravitational eld, quantum elds with their combinatorial

generalization, also with discussions on fundamental questions in epistemology. All

of these are valuable for researchers in combinatorics, topology, differential geom-

etry, gravitational or quantum elds.

7815999 731001

ISBN 1-59973-100-2

53995>

Combinatorial Geometry with Applications to Field Theory, by L. Mao

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