BENZ, Walter (2012). Classical Geometries in Modern Contexts

8/10/2019 BENZ, Walter (2012). Classical Geometries in Modern Contexts

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Walter Benz

Classical Geometriesin Modern Contexts

Third Edition

Geometry of Real Inner Product Spaces


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DOI 10.1007/978-3- 0 -Springer Basel Heidelberg New York Dordrecht London

Printed on acid-free paper

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ISBN 978-3-0348-0419-6 ISBN 978-3-0348-0420-2 (eBook)0348- 420 2

FB MathematikMathematisches Seminar

Germany

Walter Benz

Universitt Hamburg

Hamburg

Springer Basel AG is part of Springer Science+Business Media (www.birkhauser-science.com)

Springer Basel 2012

Library of Congress Control Number: 2012945178
http://www.birkhauser-science.com/


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Contents

Preface ix

Preface to the Second Edition xiii

Preface to the Third Edition xv

1 Translation Groups 1

1.1 Real inner product spaces . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Isomorphic, non-isomorphic spaces . . . . . . . . . . . . . . . . . . 3

1.4 Inequality of CauchySchwarz . . . . . . . . . . . . . . . . . . . . . 41.5 Orthogonal mappings . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 A characterization of orthogonal mappings . . . . . . . . . . . . . . 71.7 Translation groups, axis, kernel . . . . . . . . . . . . . . . . . . . . 101.8 Separable translation groups . . . . . . . . . . . . . . . . . . . . . . 141.9 Geometry of a group of permutations . . . . . . . . . . . . . . . . . 161.10 Euclidean, hyperbolic geometry . . . . . . . . . . . . . . . . . . . . 201.11 A common characterization . . . . . . . . . . . . . . . . . . . . . . 211.12 Other directions, a counterexample . . . . . . . . . . . . . . . . . . 34

2 Euclidean and Hyperbolic Geometry 37

2.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 The lines of L.M. Blumenthal . . . . . . . . . . . . . . . . . . . . . 382.3 The lines of Karl Menger . . . . . . . . . . . . . . . . . . . . . . . 432.4 Another definition of lines . . . . . . . . . . . . . . . . . . . . . . . 452.5 Balls, hyperplanes, subspaces . . . . . . . . . . . . . . . . . . . . . 462.6 A special quasihyperplane . . . . . . . . . . . . . . . . . . . . . . 502.7 Orthogonality, equidistant surfaces . . . . . . . . . . . . . . . . . . 51

2.8 A parametric representation . . . . . . . . . . . . . . . . . . . . . . 542.9 Ends, parallelity, measures of angles . . . . . . . . . . . . . . . . . 562.10 Angles of parallelism, horocycles . . . . . . . . . . . . . . . . . . . 612.11 Geometrical subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 63

v


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2.12 The CayleyKlein model . . . . . . . . . . . . . . . . . . . . . . . . 662.13 Hyperplanes under translations . . . . . . . . . . . . . . . . . . . . 702.14 Lines under translations . . . . . . . . . . . . . . . . . . . . . . . . 722.15 Hyperbolic coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 742.16 All isometries of (X, eucl), (X, hyp) . . . . . . . . . . . . . . . . . 752.17 Isometries preserving a direction . . . . . . . . . . . . . . . . . . . 772.18 A characterization of translations . . . . . . . . . . . . . . . . . . . 782.19 Different representations of isometries . . . . . . . . . . . . . . . . 792.20 A characterization of isometries . . . . . . . . . . . . . . . . . . . . 802.21 A counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.22 An extension problem . . . . . . . . . . . . . . . . . . . . . . . . . 862.23 A mapping which cannot be extended . . . . . . . . . . . . . . . . 91

3 Sphere Geometries of Mobius and Lie 933.1 Mobius balls, inversions . . . . . . . . . . . . . . . . . . . . . . . . 933.2 An application to integral equations . . . . . . . . . . . . . . . . . 963.3 A fundamental theorem . . . . . . . . . . . . . . . . . . . . . . . . 983.4 Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.5 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.6 Mobius circles,MN- and M

N-spheres . . . . . . . . . . . . . . . . 1113.7 Stereographic projection . . . . . . . . . . . . . . . . . . . . . . . . 1203.8 Poincares model of hyperbolic geometry . . . . . . . . . . . . . . . 123

3.9 Spears, Laguerre cycles, contact . . . . . . . . . . . . . . . . . . . . 1333.10 Separation, cyclographic projection . . . . . . . . . . . . . . . . . . 1393.11 Pencils and bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 1443.12 Lie cycles, groups Lie (X), Lag (X) . . . . . . . . . . . . . . . . . 1503.13 Lie cycle coordinates, Lie quadric . . . . . . . . . . . . . . . . . . . 1543.14 Lorentz boosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1593.15 M (X) as part of Lie (X) . . . . . . . . . . . . . . . . . . . . . . . 1673.16 A characterization of Lag (X) . . . . . . . . . . . . . . . . . . . . . 1703.17 Characterization of the Lorentz group . . . . . . . . . . . . . . . . 172

3.18 Another fundamental theorem . . . . . . . . . . . . . . . . . . . . . 173

4 Lorentz Transformations 175

4.1 Two characterization theorems . . . . . . . . . . . . . . . . . . . . 1754.2 Causal automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 1774.3 Relativistic addition . . . . . . . . . . . . . . . . . . . . . . . . . . 1814.4 Lightlike, timelike, spacelike lines . . . . . . . . . . . . . . . . . . . 1844.5 Light cones, lightlike hyperplanes . . . . . . . . . . . . . . . . . . . 1864.6 Characterization of some classes of hyperplanes . . . . . . . . . . . 191

4.7 L (Z) as subgroup of Lie (X) . . . . . . . . . . . . . . . . . . . . . 1934.8 A characterization of LM-distances . . . . . . . . . . . . . . . . . . 1944.9 Einsteins cylindrical world . . . . . . . . . . . . . . . . . . . . . . 1974.10 Lines, nulllines, subspaces . . . . . . . . . . . . . . . . . . . . . . 200


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4.11 2-point invariants of

C(Z), M C(Z)

. . . . . . . . . . . . . . . . 2024.12 De Sitters world . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2054.13 2-point invariants of

(Z), M (Z)

. . . . . . . . . . . . . . . . 205

4.14 Elliptic and spherical distances . . . . . . . . . . . . . . . . . . . . 2084.15 Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2104.16 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2124.17 Distance functions ofX0 . . . . . . . . . . . . . . . . . . . . . . . . 2154.18 Subspaces, balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2174.19 Periodic lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2184.20 Hyperbolic geometry revisited . . . . . . . . . . . . . . . . . . . . . 222

5 Projective Mappings, Isomorphism Theorems 231

5.1 linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

5.2 All affine mappings of (X, ) . . . . . . . . . . . . . . . . . . . . 2335.3 projective hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . 2355.4 Extensions ofaffine mappings . . . . . . . . . . . . . . . . . . . . 2365.5 All projective mappings . . . . . . . . . . . . . . . . . . . . . . . 2395.6 dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2405.7 The projective CayleyKlein model . . . . . . . . . . . . . . . . 2425.8 Mtransformations from X onto V . . . . . . . . . . . . . . . . . 2475.9 Isomorphic Mobius sphere geometries . . . . . . . . . . . . . . . . . 2495.10 Isomorphic Euclidean geometries . . . . . . . . . . . . . . . . . . . 252

5.11 Isomorphic hyperbolic geometries . . . . . . . . . . . . . . . . . . . 2565.12 A mixed case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

6 Planes of Leibniz, Lines of Weierstrass, Varia 265

6.1 Lhyperplanes of metric spaces . . . . . . . . . . . . . . . . . . . . 2656.2 Hyperbolic hyperplanes asLhyperplanes . . . . . . . . . . . . . . 2676.3 Elliptic and spherical hyperplanes . . . . . . . . . . . . . . . . . . . 2706.4 Hyperplanes of LorentzMinkowski geometry . . . . . . . . . . . . 2726.5 De Sitters world as substructure of (Z,L(Z)) . . . . . . . . . . . . 273

6.6 Hyperplanes of de Sitters world . . . . . . . . . . . . . . . . . . . 2756.7 Hyperbolic lines as defined by K. Weierstrass . . . . . . . . . . . . 2796.8 Orthogonal lines in metric spaces . . . . . . . . . . . . . . . . . . . 2826.9 Lines orthogonal to hyperplanes . . . . . . . . . . . . . . . . . . . . 2866.10 A fundamental representation of motions . . . . . . . . . . . . . . 2866.11 Dimensionfree hyperbolic geometry under mild hypotheses . . . . 2896.12 All 2point invariants of(P, G) . . . . . . . . . . . . . . . . . . . . 293

A Notation and symbols 297

B Bibliography 299

Index 305


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Preface

The basic structure playing the key role in this book is a real inner product space

(X, ), i.e. a real vector space X together with a mapping : XX R

, aso-called inner product, satisfying rules (i), (ii), (iii), (iv) of section 1 of chapter1. In order to avoid uninteresting cases from the point of view of geometry, wewill assume throughout the whole book that there exist elements a, b in X whichare linearly independent. But, on the other hand, we do not ask for the exis-tence of a positive integer n such that every subset S ofX containing exactly nelements is linearly dependent. In other words, we donotassume thatXis a finite-dimensional vector space. So, when dealing in this book with different geometrieslike euclidean, hyperbolic, elliptic, spherical, LorentzMinkowskian geometry orMobius (Lie) sphere geometry over a real inner product space (X, ), the readermight think ofX= R2 or R3, ofXfinite-dimensional, or ofXinfinite-dimensional.In fact, it plays no role, whatsoever, in our considerations whether the dimensionofX is finite or infinite: the theory as presented does not depend on the dimensionofX. In this sense we may say that our presentation in question isdimension-free.

The prerequisites for a fruitful reading of this book are essentially basedon the sophomore level, especially after mastering basic linear algebra and basicgeometry ofR2 and R3. Of course, hyperspheres are defined via the inner prod-uct . At the same time we also define hyperplanes by this product, namely by

{x X |(a, x) =}, or, as we prefer to write{x X |ax = }, with 0=a Xand R. This is a quite natural and simple definition and familiar to everybodywho learned geometry, say, of the plane or ofR3. For us it means that we do notneed to speak about the existence of a basis ofX(see, however, section 2.6 wherewe describe an example of a quasi-hyperplane which is not a hyperplane) and,furthermore, that we do not need to speak about (affine) hyperplanes as imagesunder translations of maximal subspaces=X ofX(see R. Baer [1], p. 19), henceavoiding transfinite methods, which could be considered as somewhat strange inthe context of geometries of Kleins Erlangen programme. This programme was

published in 1872 by Felix Klein (18491925) under the title Vergleichende Be-trachtungen uber neuere geometrische Forschungen, Programm zum Eintritt in diephilosophische Facultat und den Senat der k. Friedrich-Alexander-Universitat zuErlangen(Verlag von Andreas Deichert, Erlangen), and it gave rise to an ingenious

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and fundamental principle that allows distinguishing between different geometries(S, G) (see section 9 of chapter 1) on the basis of their groups G, their invariantsand invariant notions (section 9). In connection with Kleins Erlangen programmecompare also Julian Lowell Coolidge, A History of Geometrical Methods, Claren-don Press, Oxford, 1940, and, for instance, W. Benz [3], p. 38 f.

The papers [1] and [5] of E.M. Schroder must be considered as pioneer workfor a dimension-free presentation of geometry. In [1], for instance, E.M. Schroderproved for arbitrary-dimensional X, dim X 2, that a mapping f : X Xsatisfying f (0) = 0 and x1x2 = f(x1) f(x2) for all x1, x2 X withx1 x2= 1 or 2 must be orthogonal. The methods of this result turned out tobe important for certain other results of dimension-free geometry (see Theorem 4of chapter 1 of the present book, see also W. Benz, H. Berens [1] or F. Rado, D.

Andreescu, D. Valcan [1]).The main result of chapter 1 is a common characterization of euclidean and

hyperbolic geometry over (X, ). With an implicit notion of a (separable) transla-tion group T of X with axis e X(see sections 7, 8 of chapter 1) the followingtheorem is proved (Theorem 7). Let d be a function, not identically zero, fromX X into the set R0 of all non-negative real numbers satisfying d (x, y) =d

(x), (y)

and, moreover, d (e, 0) = d (0, e) = d (0, e) + d (e,e) for allx, y X, all T O (X) where O (X) is the group of orthogonal bijections ofX, and for all real , with 0 . Then, up to isomorphism, there exist

exactly two geometries with distance functiondin question, namely the euclideanor the hyperbolic geometry over (X, ). We would like to stress the fact that thisresult, the proof of which covers several pages, is also dimension-free, i.e. that itcharacterizes classical euclidean and classical (non-euclidean) hyperbolic geometrywithout restriction on the (finite or infinite) dimension ofX, provided dim X2.Hyperbolic geometry of the plane was discovered by J. Bolyai (18021860), C.F.Gau (17771855), and N. Lobachevski (17931856) by denying the euclidean par-allel axiom. In our Theorem 7 in question it is not a weakened axiom ofparallelity,but a weakened notion oftranslation with a fixed axiswhich leads inescapably to

euclidean or hyperbolic geometry and this for all dimensions ofXwith dim X2.The methods of the proof of Theorem 7 depend heavily on the theory of functionalequations. However, all results which are needed with respect to functional equa-tions are proved in the book. Concerning monographs on functional equations seeJ. Aczel [1] and J. AczelJ. Dhombres [1].

In chapter 2 the two metric spaces (X, eucl) (euclidean metric space) and(X, hyp) (hyperbolic metric space) are introduced depending on the different dis-tance functions eucl (x, y), hyp (x, y) (x, y X) of euclidean, hyperbolic geometry,respectively. The lines of these metric spaces are characterized in three different

ways, as lines of L.M. Blumenthal (section 2), as lines of Karl Menger (section 3), oras follows (section 4): for given a =b ofXcollect asline througha, ball p Xsuchthat the system d (a, p) =d (a, x) and d (b, p) =d (b, x) of two equations has onlythe solution x= p. Moreover, subspaces of the metric spaces in question are defined


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Preface xi

in chapter 2, as well as spherical subspaces, parallelism, orthogonality, angles, mea-sures of angles and, furthermore, with respect to (X, hyp), equidistant surfaces,ends, horocycles, and angles of parallelism. As far as isometries of (X, hyp) areconcerned, we would like to mention the following main result (Theorem 35, chap-ter 2) which corresponds to Theorem 4 in chapter 1. Let > 0 be a fixed realnumber andN >1 be a fixed integer. Iff :XXsatisfies hyp

f(x), f(y)

for all x, y Xwith hyp (x, y) =, and hyp

f(x), f(y)

N for all x, yXwith hyp (x, y) = N , then f must be an isometry of (X, hyp), i.e. satisfieshyp

f(x), f(y)

= hyp(x, y) for all x, y X. If the dimension ofXis finite, the

theorem of B. Farrahi [1] and A.V. Kuzminyh [1] holds true: let >0 be a fixedreal number and f : X X a mapping satisfying hyp

f(x), f(y)

= for all

x, y Xwith hyp (x, y) = . Then fmust already be an isometry. In section 21of chapter 2 an example shows that this cannot generally be carried over to theinfinite-dimensional case.

A geometry = (S, G) is a set S= together with a group G of bijectionsofSwith the usual multiplication (f g)(x) =f

g (x)

for all x S and f, gG.

The geometer then studies invariants and invariant notions of (S, G) (see section9 of chapter 1). If a geometry is based on an arbitrary real inner product spaceX, dim X 2, then it is useful, as we already realized before, to understand by, dimension-free a theory of which applies to every described X, no matterwhether finite- or infinite-dimensional, so, for instance, the same way to R2 as

to C[0, 1] with f g =10 t2f(t) g (t) dt for real-valued functions f, g defined and

continuous in [0, 1] (see section 2, chapter 1). In chapter 3 we develop the geom-etry of Mobius dimension-free, and also the sphere geometry of Sophus Lie. EvenPoincares model of hyperbolic geometry can be established dimension-free (seesection 8 of chapter 3). In order to stress the fact that those and other theories aredeveloped dimension-free, we avoided drawings in the book: drawings, of course,often present properly geometrical situations, but not, for instance, convincinglythe ball B (c, 1) (see section 4 of chapter 2) of the above mentioned example withX=C[0, 1] such thatc : [0, 1] R is the functionc () =3. The close connection

between Lorentz transformations (see section 17 of chapter 3) and Lie transfor-mations (section 12), more precisely Laguerre transformations (section 13), hasbeen known for almost a hundred years: it was discovered by H. Bateman [1] andH.E. Timerding [1], of course, in the classical context of four dimensions (section17). This close connection can also be established dimension-free, as shown inchapter 3. A fundamental theorem in LorentzMinkowski geometry (see section17, chapter 3) of A.D. Alexandrov [1] must be mentioned here with respect to Liesphere geometry: if (2)dim X


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the Cacciafesta form (Cacciafesta [1]) (see Theorem 2 of chapter 4).

All Lorentz transformations of LorentzMinkowski geometry over (X, ) aredetermined dimension-free in chapter 4, section 1, by Lorentz boosts (section 14,

chapter 3), orthogonal mappings and translations. Also this result follows from atheorem (Theorem 61 in section 14, chapter 3) on Lie transformations. In The-orem 6 (section 2, chapter 4) we prove dimension-free a well-known theorem ofAlexandrovOvchinnikovaZeeman which these authors have shown under the as-sumption dim X


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Preface to the Second Edition

In this second edition a new chapter (Projective Mappings, Isomorphism The-orems) was added. One of the fundamental results contained in this chapter 5 isthat the hyperbolic geometries over two (not necessarily finitedimensional) realinner product spaces (X, ), (V, ) (see p. 1) are isomorphic (p. 16f) if, and only if,the two underlying real inner product spaces are isomorphic (p. 1f) as well. Simi-lar theorems are proved for Mobius sphere geometries and for the euclidean case.Another result of chapter 5 we would like to mention is that the CayleyKleinmodel of hyperbolic geometry over (X, ), as developed dimensionfree in section2.12, can also be established dimensionfree via a certain selection of projectivemappings ofXdepending, however, on the chosen inner product ofX.

It remains to the author to thank Professors Hans Havlicek, Zsolt Pales, Vic-tor Pambuccian who, through their support, their criticism and their suggestions,contributed to the improvement of this book. Special thanks in this connectionare due to Alice Gunther and my colleagues Ludwig Reich and Jens Schwaiger.

Last, but not least, I would like to express my gratitude to the Birkhauserpublishing company and, especially, to Dr. Thomas Hempfling for their conscien-tious work and helpful cooperation.

Hamburg, July 2007 Walter Benz

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Preface to the Third Edition

During the first decades of the 20th century, Geometry consisted of theories ofthe plane and threedimensional space, with exceptions, of course, such as LiesSphere Geometry, Pluckers Line Geometry, and Einsteins Special Theory of Rel-ativity. These theories concerned, for instance, Geometries of Kleins ErlangenProgramme, Hilberts Foundations of Geometry, Higher Geometry, and Differ-ential Geometry. Subsequently, mathematicians began to study more intensivelyndimensional Geometry, for n an integer 2, however, based on approaches ofA. Cayley (18211895), H.G. Grassmann (18091877). Some of the major resultsof this research were most certainly the modern theory of Linear Algebra, the newindispensable tools of general vector spaces and of real inner product spaces. Theclassical inner product itself, say ofR3, soon became generalized, because of its

fundamental importance in both Geometry and Analysis.Looking to the examples X(B) in a), p. 2, of real inner product spaces, one

is astonished at how many such spaces X(B) exist, with B = an arbitrary set,taking into account (see p. 3) that X(B1), X(B2) are isomorphic if, and only if,there exists a bijection betweenB1 and B2. The many existing real inner productspaces, including the countable set of all Rn, n= 2, 3, . . ., and many other interest-ing structures (see section 1.2), certainly deserve their own geometrical treatment,namely a Geometry of Real Inner Product Spaces, as have developed in this book.

Also this should be emphasized: an important postulate of the 1950s, es-sentially to avoid coordinates in Geometry is definitely realized when working,without referring to a basis, with elements of arbitrary real inner product spaces.

Further developments in the Geometry of Real Inner Product Spaces areinevitably important, an idea even now promoted by the fact that a third editionof the book became necessary.

In this third edition a new chapter (6. Planes of Leibniz, Lines of Weierstrass,Varia) was added. One of the fundamental results proved in this chapter concernsthe representation of hyperbolic motions: every M(X, hyp) can be written

in the form = T with a uniquely determined hyperbolic translation T anda uniquely determined bijective orthogonal transformation (Theorem 13). Westress the fact that this holds true for all real inner product spaces Xof arbitrary(finite or infinite) dimension> 1. Observe, however, that the set of all translations

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ofXis not a subgroup of the hyperbolic group ofX.

Another fundamental result of chapter 6 concerns the Geometry (P, G) ofsegments. Let X be a real inner product space of arbitrary (finite or infinite)

dimension>1. Define P :={x X | x< 1}and Gto be the group of all bijec-tions ofPsuch that the images ofPlines are again Plines. Then the Geometry(P, G) of segments is isomorphic to the hyperbolic geometry over X (Theorem16). What one usually learns in a basic course in Geometry is formulated, say forR

3, that if is the set of points

:={(x1, x2, x3) R3 |x21+ x

2

2+ x2

3


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Dedicated to Janos Aczel


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BENZ, Walter (2012). Classical Geometries in Modern Contexts

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