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6

Riemannian Geometry of Noncommutative Surfaces

M. Chaichiana1, A. Tureanua2, R. B. Zhangb3 and X. Zhangc4

aHigh Energy Physics Division, Department of Physical Sciences, University of Helsinki,and Helsinki Institute of Physics, P.O. Box 64, 00014 Helsinki, Finland

b School of Mathematics and Statistics, University of Sydney, Sydney, AustraliacInstitute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of

Sciences, Beijing, China

1. INTRODUCTION

In recent years there has been much progress in developing theories of noncommutative geometryand exploring their applications in physics. Many viewpoints were adopted and different mathematicalapproaches were followed by different researchers. Connes’ theory [10] (see also [20]) formulatedwithin the framework ofC∗-algebras is the most successful approach to noncommutative differentialgeometry. It incorporates cyclic cohomology and K-theory,and gives rise to noncommutative versionsof index theorems. Theories generalizing aspects of algebraic geometry were also developed (see, e.g.,[26] for a review and references). A notion of noncommutative schemes was formulated, which seemsto provide a useful framework for developing noncommutative algebraic geometry.

A major advance in theoretical physics in recent years was the deformation quantization of Poissonmanifolds by Kontsevich (see [22] for the final form of this work). This sparked intensive activitiesinvestigating applications of noncommutative geometriesto quantum theory. The work of Seiberg andWitten [25] showed that the anti-symmetric tensor field arising from massless states of strings can bedescribed by the noncommutativity of a spacetime,

(1.1) [xµ,xν]∗ = iθµν, θµν constant matrix,

where the multiplication of the algebra of functions is governed by the Moyal product

(1.2) ( f ∗g)(x) = f (x)exp

(

i2

θµν←−∂ µ−→∂ ν

)

g(x) .

A considerable amount of research was done both prior and after [25], and we refer to [28, 16] forreviews and references.

An earlier and independent work is the seminal paper [14] by Doplicher, Fredenhagen and Roberts,which laid down the fundamentals of quantum field theory on noncommutative spacetime. These au-thors started with a theoretical examination of the long held belief by the physics community that theusual notion of spacetime needed to be modified at the Planck scale, and convincingly demonstratedthat spacetime becomes noncommutative in that the coordinates describing spacetime points becomeoperators similar to those in quantum mechanics. Therefore, noncommutative geometry is indeed a wayto describe physics at Planck scale.

A consistent formulation of a noncommutative version of general relativity could give an insight intoa gravitational theory compatible with quantum mechanics.Although a unification of general relativity

1Masud.Chaichian@helsinki.fi2Anca.Tureanu@helsinki.fi3rzhang@maths.usyd.edu.au4xzhang@amss.ac.cn

1

2

with quantum mechanics has long been sought after but remains as elusive as ever despite the extraor-dinary progress in string theory for the last two decades. The noncommutative geometrical approachto gravity could provide an alternative route. Much work hasalready been done in this general direc-tion, see, e.g., [8, 23, 24, 2, 3] and references therein. In particular, different forms of noncommutativeRiemannian geometries were developed [23, 24, 2], which retain some of the familiar geometric no-tions like metric and curvature. Noncommutative analoguesof the Hilbert-Einstein action were alsosuggested [8, 9, 2] by treating noncommutative gravity as gauge theories.

The noncommutative spacetime with the Heisenberg-like commutation relation (1.1) violates Lorentzsymmetry but retains translational invariance. It was shown that noncommutative field theory formulatedon such a spacetime actually has a quantum symmetry under thetwisted Poincare algebra [4]. TheAbelian twist element

(1.3) F = exp

(

− i2

θµν∂µ⊗ ∂ν

)

was used in [4] to twist the universal enveloping algebra of the global Poincare algebra, obtaining theMoyal star-product (1.2) for the algebra of functions on thePoincare group. It is then natural to tryto extend the procedure to other symmetries of noncommutative field theory and investigate whetherthe concept of twist provides a new symmetry principle for noncommutative spacetime. In an attemptto construct a noncommutative gravitational theory, the same Abelian twist element (1.3) was usedin [2] for deforming the algebra of diffeomorphisms, with the hope of obtaining general coordinatetransformations on the noncommutative space-time. However, the resulting gravitational theory is notcompatible with the low-energy limit of string theory [1]. This is not at all surprising. Even thoughthe procedure of twisting the algebra of diffeomorphisms isin principle desirable, the choice of theframe-dependent twist element (1.3) causes inconsistencies since physically the deformation of generalcoordinate transformations must be done in a frame-independent manner. Basic physical argumentsrequire that the frame-dependent Moyal product transformsunder a general coordinate transformation;however, if the twist element is chosen as (1.3), the Moyal∗-product is fixed once for all by the choiceof the twist and does not transform. Technically the inconsistencies, which such a choice leads to, aresimilar to those shown [5] to result when one attempted to deform the internal gauge transformationswith the same twist element (1.3).

Nevertheless twisting is expected to be a productive approach to the formulation of a noncommuta-tive gravitational theory if implemented in a consistent manner. A “covariant twist” was proposed forinternal gaugetransformations in [6], but it turned out that the corresponding star-product would notbe associative. This purely algebraic approach has not yet been studied in the case ofgauge Poincaretransformations.

The present paper aims at the modest goal of constructing concrete examples of noncommutativeRiemannian geometries, which should be simple and transparently consistent. Our approach is math-ematically different from that of [2] and also quite far removed from the quantum group theoreticalnoncommutative Riemannian geometry of [24] (see also references therein and subsequent publicationsby the same author).

Recall that 2-dimensional surfaces embedded in the Euclidean 3-space provide the simplest yet non-trivial examples of Riemannian geometry. The Euclidean metric of the 3-space induces a natural metricfor a surface through the embedding; the Levi-Civita connection and the curvature of the tangent bundleof the surface can thus be described explicitly (for the theory of surfaces, see, e.g., the textbook [15]).

We begin by developing in Section 2 a noncommutative Riemannian geometry for noncommutativeanalogues of 2-dimensional surfaces embedded in 3-space. We work over an associative algebraA,which is a deformation [18] of the algebra of smooth functions on a region ofR2 and show that much ofthe classical differential geometry for surfaces generalizesnaturallyto this noncommutative setting. Weemphasize that the embeddings play a crucial role in understanding the geometry of the noncommutative

3

surfaces. The illuminating examples of the noncommutativesphere and torus are discussed in this spiritin Section 3.

Once the noncommutative Riemannian geometry of the 2-dimensional surfaces is sorted out, itsgeneralization to the noncommutative geometries corresponding ton-dimensional surfaces embeddedin spaces of higher dimensions is straightforward. This is discussed in Section 4.

Recall that the basic principle of general relativity is general covariance. We study in Section 5general coordinate transformations for noncommutative surfaces, which are brought about by gaugetransformations on the underlying noncommutative associative algebraA (over which noncommutativegeometry is constructed). A new feature here is that the general coordinate transformations transform themultiplication of the underlying associative algebraA as well, turning it into another algebra nontriviallyisomorphic toA. We make comparison with classical Riemannian geometry, showing that the gaugetransformations should be considered as noncommutative analogues of diffeomorphisms.

The theory of surfaces developed over the deformation of thealgebra of smooth functions on someregion inR

n now suggests a general theory of noncommutative Riemanniangeometry ofn-dimensionalsurfaces over arbitrary unital associative algebras with derivations. We present a brief outline of thisgeneral theory in Section 6.

We conclude this section with a remark on the presentation ofthe paper. As indicated above, we startfrom the simplest nontrivial examples of noncommutative Riemannian geometries and gradually extendthe results to build up a theory of generality. This “experimental approach” is not the optimal format forpresenting mathematics, as all special cases repeat the same pattern. However, it has the advantage thatthe general theory obtained in this way stands on a firm ground.

2. NONCOMMUTATIVE SURFACES AND THEIR EMBEDDINGS

2.1. Noncommutative surfaces and their embeddings.We fix a regionU in R2, and write the coor-

dinate of a pointt in U as(t1, t2). Let h be a real indeterminate, and denote byR[[h]] the ring of formalpower series inh. Let A be the set of the formal power series inh with coefficients being real smoothfunctions onU . Namely, every element ofA is of the form∑i≥0 fi hi where fi are smooth functions onU . ThenA is anR[[h]]-module in the obvious way.

Given any two smooth functionsf andg onU , we denote byf g the usual point-wise product of thetwo functions. We also define their star-product (or more precisely, Moyal product) by

f ∗g = limt′→t

exp

[

h

(

∂∂t1

∂∂t ′2− ∂

∂t2

∂∂t ′1

)]

f (t)g(t ′),(2.1)

where the exponential exp[h( ∂∂t1

∂∂t′2− ∂

∂t2∂

∂t′1)] is to be understood as a power series in the differential

operator ∂∂t1

∂∂t′2− ∂

∂t2∂

∂t′1. More explicitly, let

µp : A/hA⊗A/hA−→A/hA, p = 0,1,2, . . . ,(2.2)

beR-linear maps defined by

µp( f ,g) = limt′→t

1p!

(

∂∂t1

∂∂t ′2− ∂

∂t2

∂∂t ′1

)p

f (t)g(t ′).

Then f ∗g= ∑∞p=0 hpµp( f ,g). It is evident thatf ∗g lies inA. We extend this star-productR[[h]]-linearly

to all elements inA by letting

(∑ fi hi)∗ (∑g j h

j) := ∑ fi ∗g j hi+ j .

It has been known since the early days of quantum mechanics that the Moyal product is associative (see,e.g., [22] for a reference), thus we arrive at an associativealgebra overR[[h]], which is a deformation

4

[18] of the algebra of smooth functions onU . We shall usually denote this associative algebra byA, butwhen it is necessary to make explicit the multiplication of the algebra, we shall write it as(A,∗).Remark2.1. For the sake of being explicit, we restrict ourselves to consider the Moyal product (definedby (2.1)) only in this section. As we shall see in Sections 5 and 6, the theory of noncommutative surfacesto be developed in this paper extends to more general star-products over algebras of smooth functions.

Write ∂i for ∂∂ti

, and extendR[[h]]-linearly the operators∂i to A. One can easily verify that for smoothfunctions f andg,

∂i( f ∗g) = (∂i f )∗g+ f ∗ (∂ig),(2.3)

that is, the operators∂i are derivations of the algebraA.Let A3 = A⊕A⊕A. There is a natural two-sidedA-module structure onA3⊗

R[[h]] A3, defined for

all a,b∈A, andX⊗Y ∈A3⊗R[[h]] A

3 by a∗ (X⊗Y)∗b = a∗X⊗Y∗b. Define the map

A3⊗

R[[h]] A3−→A, (a,b,c)⊗ ( f ,g,h) 7→ a∗ f +b∗g+c∗h,(2.4)

and denote it by•. This is a map of two-sidedA-modules in the sense that for anyX,Y ∈ A3 anda,b∈A, (a∗X)• (Y∗b) = a∗ (X •Y)∗b. We shall refer to this map as thedot-product.

LetX = (X1,X2,X3) be an element ofA3, where the superscripts ofX1, X2 andX3 are not powers butare indices used to label the components of a vector as in the usual convention in differential geometry.We set∂iX = (∂iX1,∂iX2,∂iX3), and define the following 2×2-matrix overA

g =

(

g11 g12

g21 g22

)

, gi j = ∂iX • ∂ jX.(2.5)

Let g0 = g modh, which is a 2×2-matrix of smooth functions onU .

Definition 2.2. We call an elementX ∈A3 (thenoncommutative embeddingin A3 of) anoncommutativesurfaceif g0 is invertible for allt ∈U . In this case, we callg themetricof the noncommutative surface.

Given a noncommutative surfaceX with a metricg, there exists a unique 2×2-matrix(

gi j)

overAwhich is the right inverse ofg, i.e.,

gi j ∗g jk = δki ,

where we have used Einstein’s convention of summing over repeated indices. To see the existence of theright inverse, we writegi j = ∑p hpgi j [p] andgi j = ∑p hpgi j [p], where(gi j [0]) is the inverse of(gi j [0]).Now in terms of the mapsµk defined by (2.2), we have

δki = gi j ∗g jk = ∑

qhq ∑

m+n+p=qµp(gi j [m],g jk[n]),

which leads to

gi j [q] = −q

∑n=1

q−n

∑m=0

gik[0]µn(gkl [m], gl j [q−n−m]).

Since the right-hand side involves onlygl j [r] with r < q, this equation gives a recursive formula for theright inverse ofg.

In the same way, we can also show that there also exists a unique left inverse ofg. It follows from theassociativity of multiplication of matrices over any associative algebra that the left and right inverses ofg are equal.

Definition 2.3. Given a noncommutative surfaceX, let

Ei = ∂iX, i = 1,2,

5

and call the leftA-moduleTX and rightA-moduleTX defined by

TX = {a∗E1+b∗E2 | a,b∈A}, TX = {E1∗a+E2∗b | a,b∈A}the left andright tangent bundlesof the noncommutative surface respectively.

ThenTX⊗R[[h]] TX is a two-sidedA-module.

Proposition 2.4. The metric induces a homomorphism of two-sidedA-modules

g : TX⊗R[[h]] TX−→A,

defined for any Z= zi ∗Ei ∈ TX and W= Ei ∗wi ∈ TX by

Z⊗W 7→ g(Z,W) = zi ∗gi j ∗wj .

It is easy to see that the map is indeed a homomorphism of two-sided A-modules, and it clearlycoincides with the restriction of the dot-product toTX⊗

R[[h]] TX.Since the metricg is invertible, we can define

Ei = gi j ∗E j , Ei = E j ∗g ji ,(2.6)

which belong toTX andTX respectively. Then

g(Ei ,E j) = δij = g(E j , E

i), g(Ei , E j) = gi j .

Now anyY ∈A3 can be written asY = yi ∗Ei +Y⊥, with yi = Y • Ei, andY⊥ = Y−yi ∗Ei. We shallcall yi ∗Ei the left tangential componentandY⊥ the left normal componentof Y. Let

(TX)⊥ = {N ∈A3 | N•Ei = 0,∀i},which is clearly a leftA-submodule ofA3. In a similar way, we may also decomposeY into Y =Ei ∗ yi +Y⊥ with theright tangential componentofY given byyi = Ei •Y and theright normal componentby Y⊥ = Y−Ei ∗ yi . Let

(TX)⊥ = {N ∈A3 | Ei •N = 0,∀i},which is a rightA-submodule ofA3. Therefore, we have the following decompositions

A3 = TX⊕ (TX)⊥, as leftA-module,

A3 = TX⊕ (TX)⊥, as rightA-module.(2.7)

It follows that the tangent bundles are finitely generated projective modules overA. Following thegeneral philosophy of noncommutative geometry [10], we mayregard finitely generated projective mod-ules overA as vector bundles on the noncommutative surface. This justifies the terminology of left andright tangent bundles forTX andTX.

In fact TX and TX are free left and rightA-modules respectively, asE1 andE2 form A-bases forthem. ConsiderTX for example. If there exists a relationai ∗Ei = 0, whereai ∈A, we haveai ∗Ei •E j

= ai ∗gi j = 0, ∀i. The invertibility of the metric then leads toai = 0, ∀i. SinceE1 andE2 generateTX,they indeed form anA-basis ofTX.

One can introduce connections to the tangent bundles by following the standard procedure in thetheory of surfaces [15].

Definition 2.5. Define operators

∇i : TX−→ TX, i = 1,2,

by requiring that∇iZ is equal to the left tangential component of∂iZ for all Z ∈ TX. Similarly define

∇i : TX−→ TX, i = 1,2,

by requiring that∇iZ is equal to the right tangential component of∂i Z for all Z ∈ TX. Call the setconsisting of the operators∇i (respectively∇i) aconnectiononTX (respectivelyTX).

6

The following result justifies the terminology.

Lemma 2.6. For all Z ∈ TX, W∈ TX and f∈A,

∇i( f ∗Z) = ∂i f ∗Z+ f ∗∇iZ, ∇i(W ∗ f ) = W ∗ ∂i f + ∇iW∗ f .(2.8)

Proof. By the Leibniz rule (2.3) for∂i ,

∂i( f ∗Z) = (∂i f )∗Z+ f ∗ (∂iZ), ∂i(W ∗ f ) = W ∗ (∂i f )+W ∗ (∂i f ).

The lemma immediately follows from the tangential components of these relations under the decompo-sitions (2.7). �

In order to describe the connections more explicitly, we note that there existΓki j andΓk

i j in A suchthat

∇iE j = Γki j ∗Ek, ∇iE j = Ek ∗ Γk

i j .(2.9)

Because the metric is invertible, the elementsΓki j andΓk

i j are uniquely defined by equation (2.9). Wehave

Γki j = ∂iE j • Ek Γk

i j = Ek • ∂iE j .(2.10)

It is evident thatΓki j andΓk

i j are symmetric in the indicesi and j. The following closely related objectswill also be useful later:

Γi jk = ∂iE j •Ek, Γi jk = Ek • ∂iE j .

In contrast to the commutative case,Γki j andΓk

i j do not coincide in general, and they depend explicitlyon the embeddingX of the noncommutative surface. Let

cΓi jl =12

(

∂ig jl + ∂ jgli − ∂lg ji)

,

ϒi jl =12

(∂iE j •El −El • ∂iE j) ,

then we haveΓk

i j = cΓi jl ∗glk + ϒi jl ∗glk, Γki j = gkl ∗ cΓi jl +gkl ∗ϒi jl .

Note that in the classical limit withh = 0, ϒki j vanishes and bothΓk

i j and Γki j reduce to the standard

Levi-Civita connection.We have the following result.

Proposition 2.7. The connections are metric compatible in the following sense

∂ig(Z, Z) = g(∇iZ, Z)+g(Z, ∇iZ), ∀Z ∈ TX, Z ∈ TX.(2.11)

This is equivalent to the fact that

∂ig jk−Γi jk− Γik j = 0.(2.12)

Proof. Sinceg is a map of two-sidedA-modules, it suffices to prove (2.11) by verifying the special casewith Z = E j andZ = Ek. We have

∂ig(E j ,Ek) = ∂i(E j •Ek)

= ∂iE j •Ek +E j • ∂iEk

= g(∇iE j ,Ek)+g(E j , ∇iEk),

where the second equality is equivalent (2.12). This provesboth statements of the proposition. �

Remark2.8. In generalΓi jk and Γi jk are not equal, thus equation (2.12) by itself is not sufficient todetermine them uniquely in contrast to the commutative case.

7

Remark2.9. At this point we should relate to the literature. The metric introduced here resemblessimilar notions in [23, 12, 13, 11]; also our left and right connections and their metric compatibilityhave much similarity with Definitions 2 and 3 in [11]. However, there are crucial differences. Our left(respectively right) tangent bundle is a left (respectively right) A-module only, while in [12, 11] there isonly one “tangent bundle”T which is a bimodule over some algebra (or Hopf algebra)B. The metricsdefined in [23, 12, 13, 11] are maps fromT⊗B T to B. A noteworthy feature of the metric in [23] is thata particular moving frame can be chosen to make all the components of the metric central [23, (3.22)].

2.2. Curvatures and second fundamental form.Let [∇i ,∇ j ] := ∇i∇ j −∇ j∇i and[∇i , ∇ j ] := ∇i∇ j −∇ j ∇i . Straightforward calculations show that for allf ∈A,

[∇i ,∇ j ]( f ∗Z) = f ∗ [∇i ,∇ j ]Z, Z ∈ TX,

[∇i , ∇ j ](W ∗ f ) = [∇i , ∇ j ]W∗ f , W ∈ TX.

Clearly the right-hand side of the first equation belongs toTX, while that of the second equation belongsto TX. We re-state these important facts as a proposition.

Proposition 2.10. The following maps

[∇i ,∇ j ] : TX−→ TX, [∇i , ∇ j ] : TX−→ TX

are left and rightA-module homomorphisms respectively.

SinceTX (respectivelyTX) is generated byE1 andE2 as a left (respectively right)A-module, byProposition 2.10, we can always write

[∇i ,∇ j ]Ek = Rlki j ∗El , [∇i , ∇ j ]Ek = El ∗ Rl

ki j(2.13)

for someRlki j ,R

lki j ∈A.

Definition 2.11. We refer toRlki j andRl

ki j respectively as theRiemann curvaturesof the left and righttangent bundles of the noncommutative surfaceX.

The Riemann curvatures are uniquely determine by the relations (2.13). In fact, we have

Rlki j = g([∇i ,∇ j ]Ek, E

l ), Rlki j = g(El , [∇i , ∇ j ]Ek).(2.14)

Simple calculations yield the following result.

Lemma 2.12.Rl

ki j = −∂ jΓlik−Γp

ik ∗Γljp + ∂iΓl

jk + Γpjk ∗Γl

ip,

Rlki j = −∂ j Γl

ik− Γljp ∗ Γp

ik + ∂iΓljk + Γl

ip ∗ Γpjk

Proposition 2.13. Let Rlki j = Rpki j ∗gpl andRlki j =−gkp∗ Rp

li j . The Riemann curvatures of the left and

right tangent bundles coincide in the sense that Rkli j = Rkli j .

Proof. By Proposition 2.7, we haveRlki j = (∇i∇ j −∇ j∇i)Ek •El , which can be re-written as

Rlki j = ∂i(∇ jEk •El)−∇ jEk • ∇iEl

−∂ j(∇iEk •El)+ ∇iEk • ∇ jEl

= ∂i(∇ jEk •El +Ek• ∇ jEl )−Ek• ∇i∇ jEl

−∂ j(∇iEk •El +Ek• ∇iEl )+Ek• ∇ j∇iEl .

Again by Proposition 2.7, the first term on the far right-handside can be written as∂i∂ jgkl, and the thirdterm can be written as−∂i∂ jgkl. Thus they cancel out, and we arrive at

Rlki j = −Ek• (∇i∇ j − ∇ j∇i)El = Rlki j .

�

8

Because of the proposition, we only need to study the Riemannian curvature on one of the tangentbundles. Note thatRkli j = −Rkl ji , but there is no simple rule to relateRlki j to Rkli j in contrast to thecommutative case.

Definition 2.14. Let

Ri j = Rpip j , R= g ji ∗Ri j ,(2.15)

and call them the Ricci curvature and scalar curvature of thenoncommutative surface respectively.

Then obviously

Ri j =−g([∇ j ,∇l ]Ei , El ), R=−g([∇i,∇k]E

i , Ek).(2.16)

In the theory of classical surfaces, the second fundamentalform plays an important role. A similarnotion exists for noncommutative surfaces.

Definition 2.15. We define the left and rightsecond fundamental formsof the noncommutative surfaceX by

hi j = ∂iE j −Γki j ∗Ek, hi j = ∂iE j −Ek∗ Γk

i j .(2.17)

It follows from equation (2.9) that

hi j •Ek = 0, Ek • hi j = 0.(2.18)

Remark2.16. Both the left and right second fundamental forms reduce toh0i j N in the commutative limit,

whereh0i j is the standard second fundamental form andN is the unit normal vector.

The Riemann curvatureRlki j = (∇i∇ j−∇ j∇i)Ek•El can be expressed in terms of the second funda-mental forms. Note that

Rlki j = ∂ jEk • ∂iEl − ∂ jEk• ∇iEl − ∂iEk • ∂ jEl + ∂iEk • ∇ jEl .

By Definition 2.15,

Rlki j = ∂ jEk• hil − ∂iEk • h jl

= (∇ jEk +h jk)• hil − (∇iEk +hik)• h jl .

Equation (2.18) immediately leads to the following result.

Lemma 2.17. The following generalized Gauss equation holds:

Rlki j = h jk • hil −hik • h jl .(2.19)

Before closing this section, we mention that the Riemannianstructure of a noncommutative surfaceis a deformation of the classical Riemannian structure of a surface by including quantum corrections.The embedding intoA3 is not subject to any constraints as the general theory stands. However, one mayconsider particular noncommutative surfaces with embeddings satisfying extra symmetry requirementssimilar to the way in which various star products onR

3 were obtained from the Moyal product onR4 in

[19, Sections 4, 5].

3. EXAMPLES

In this section, we consider in some detail two concrete examples of noncommutative surfaces: thenoncommutative sphere and torus.

9

3.1. Noncommutative sphere.LetU = (0,π)×(0,2π), and we writeθ andφ for t1 andt2 respectively.Let X(θ,φ) = (X1(θ,φ),X2(θ,φ),X3(θ,φ)) be given by

X(θ,φ) =

(

sinθcosφcoshh

,sinθsinφ

coshh,

√cosh2hcosθ

coshh

)

(3.1)

with the components being smooth functions in(θ,φ) ∈U . It can be shown thatX satisfies the followingrelation

X1∗X1+X2∗X2+X3∗X3 = 1.(3.2)

Thus we may regard the noncommutative surface defined byX as an analogue of the sphereS2. Weshall denote it byS2

hand refer to it as anoncommutative sphere. We have

E1 =

(

cosθcosφcoshh

,cosθsinφ

coshh,−√

cosh2hsinθcoshh

)

,

E2 =

(

−sinθsinφcoshh

,sinθcosφ

coshh,0

)

.

The componentsgi j = Ei •E j of the metricg onS2h

can now be calculated, and we obtain

g11 = 1, g22 = sin2 θ− sinh2 h

cosh2 hcos2 θ,

g12 =−g21 =sinhh

coshh

(

sin2 θ−cos2 θ)

.

The components of this metric commute with one another as they depend onθ only. Thus it makessense to consider the usual determinantG of g. We have

G =sin2 θ+ tanh2 h(cos22θ−cos2 θ)

=sin2 θ[1+ tanh2 h(1−4cos2 θ)].

The inverse metric is given by

g11 =sin2 θ− tanh2 hcos2 θ

sin2 θ+ tanh2 h(cos22θ−cos2 θ),

g22 =1

sin2 θ+ tanh2 h(cos22θ−cos2 θ),

g12 =−g21 =tanhhcos2θ

sin2 θ+ tanh2 h(cos22θ−cos2 θ).

Now we determine the curvature tensor of the noncommutativesphere. The computations are quitelengthy, thus we only record the results here. For the Christoffel symbols, we have

Γ111 =−Γ111 = tanhhsin2θ, Γ122 = Γ122 =sin2θ

2cosh2h,

Γ212 = Γ212 =sin2θ

2cosh2h, Γ221 = Γ221 =− sin2θ

2cosh2h,

rest= 0,

10

Note thatΓ111 6= Γ111 (c.f. Remark 2.8). Since onlyΓki j are needed for the computation of the curvatures

beside the metric, we consider them only here. We have

Γ111 =

tanhhsin2θ(sin2 θ− tanh2 hcos2 θ)

G, Γ2

11 =tanh2 hsin2θcos2θ

G,

Γ112 = Γ1

21 =− tanhhsin2θcos2θ2cosh2 hG

, Γ212 = Γ2

21 =sin2θ

2cosh2 hG,

Γ122 =−sin2θ(sin2 θ+ tanh2 hcos2 θ)

2cosh2 hG, Γ2

22 =− tanhhsin2θcos2θ2cosh2 hG

.

From them we obtain

R1112=tanhhsin22θ

cosh2 hG, R2212=

tanhhsin22θ2cosh4 hG

,

R2112=− sin2θ2cosh2 h

∂1GG

+cos2θ

cosh2 hG(G− tanh2 hsin22θ)

+sin22θ

4cosh4 hG(1− tanhh+ tanhhcosh2hcos2θ),

R1212=sin2θ

2cosh2 h

∂1GG− 1

cosh2 hG[sin2 θ(cos2θ+2cos2 θ)

− tanh2 h(cos2 θ−2cos2 θcos2θ+cos4θcos2θ)]

+sin22θ

4cosh4 hG(1− tanhh+ tanhhcosh2hcos2θ).

It is useful to find the asymptotic expansions of the curvature tensors with respect toh:

R1112=4cos2 θh−4cos2 θ(1−4cos2 θ)h2 +O(h3),

R2112=−sin2 θ+4cos2 θcos2θh− (8cos4 θ−sin2 θ)h2 +O(h3),

R1212=sin2 θ+4cos2 θcos2θh− (6−7sin2 θ)h2 +O(h3),

R2212=2cos2 θh−2cos2 θ(1−4cos2 θ)h2 +O(h3).

We can also compute asymptotic expansions of the Ricci curvature tensor

R11 =1− (1−4cos2 θ+4cos2 θcos2θ

sin2 θ)h+O(h2),

R21 =− (cos2θ+2cos2 θsin2 θ

)h+O(h2),

R12 =(4cos2 θ+cos2θ)h+O(h2),

R22 =sin2 θ+[4cos2 θcos2θ−sin2 θ(1−4cos2 θ)− 2cos2 θcos2θsin2 θ

]h+O(h2),

and the scalar curvature

R= 2− [4(1−4cos2 θ)− 2cos2 θcos2θsin4 θ

]h+O(h2).

By settingh = 0, we obtain from the various curvatures ofS2h

the corresponding objects for the usualsphereS2. This is a useful check that our computations above are accurate.

Remark3.1. The noncommutative surfaceX =(

t1,t2,(1− t21− t2

2)1/2)

also satisfies (3.2), and reducesto the upper half of the usual sphere in the limith→ 0. However, the metric of the noncommutativesurface is very different from that of the usual sphere in thecommutative setting.

11

3.2. Noncommutative torus. This time we shall takeU = (0,2π)× (0,2π), and denote a point inUby (θ,φ). Let X(θ,φ) = (X1(θ,φ),X2(θ,φ),X3(θ,φ)) be given by

X(θ,φ) = ((a+sinθ)cosφ,(a+sinθ)sinφ,cosθ)(3.3)

wherea > 1 is a constant. ClassicallyX is the torus. When we extend scalars fromR to R[[h]] andimpose the star product on the algebra of smooth functions,X gives rise to a noncommutative torus,which will be denoted byT2

h. We have

E1 = (cosθcosφ,cosθsinφ,−sinθ) ,

E2 = (−(a+sinθ)sinφ,(a+sinθ)cosφ,0) .

The componentsgi j = Ei •E j of the metricg onT2h

take the form

g11 = 1+sinh2 hcos2θ,

g22 = (a+coshhsinθ)2−sinh2 hcos2 θ,

g12 =−g21 =−sinhhcoshhcos2θ+asinhhsinθ.

As they depend only onθ, the components of the metric commute with one another. The inverse metricis given by

g11 =(a+coshhsinθ)2−sinh2 hcos2 θ

G,

g22 =1+sinh2 hcos2θ

G,

g12 =−g21 =sinhhcoshhcos2θ+asinhhsinθ

G,

whereG is the usual determinant ofg given by

G = (sinθ+acoshh)2−a2sin2 θsinh2 h.

Now we determine the curvature tensor of the noncommutativetorus. The computations can becarried out in much the same way as in the case of the noncommutative sphere, and we merely recordthe results here. For the connection, we have

Γ112 = asinhhcosθ, Γ122 = (sinθ+acoshh)cosθ,

Γ212 = (sinθ+acoshh)cosθ, Γ221 =−(sinθ+acoshh)cosθ,

rest= 0,

and

Γ111 = G−1asinh2 hcosθ(asinθ−coshhcos2θ),

Γ211 = G−1asinhhcosθ(1+sinh2 hcos2θ),

Γ112 = Γ1

21 = G−1sinhhcosθ(sinθ+acoshh)(asinθ−coshhcos2θ),

Γ212 = Γ2

21 = G−1cosθ(sinθ+acoshh)(1+sinh2 hcos2 θ),

Γ122 =−G−1cosθ(sinθ+acoshh)[(a+coshhsinθ)2−sinh2 hcos2 θ],

Γ222 = G−1sinhhcosθ(sinθ+acoshh)(asinθ−coshhcos2θ).

12

We can find the asymptotic expansions of the curvature tensors with respect toh:

R1112=2cos2 θ(a+2sinθ)

a+sinθh+O(h3),

R2112=−sinθ(a+sinθ)+O(h2),

R1212=sinθ(a+sinθ)+O(h2),

R2212=−2cos2 θ(1+asinθ)h+O(h3).

We can also compute asymptotic expansions of the Ricci curvature tensor

R11 =sinθ

a+sinθ+O(h2),

R21 =− cos2θ(1+asinθ)

a+sinθh+O(h3),

R12 =3cos2θ+2+a(cos2 θ+cos2θ)

a+sinθh+O(h3),

R22 =sinθ(a+sinθ)+O(h2),

and the scalar curvature

R=2sinθ

(a+sinθ)+O(h2).

By settingh = 0, we obtain from the various curvatures ofT2h

the corresponding objects for the usualtorusT2.

4. NONCOMMUTATIVE n-DIMENSIONAL SURFACES

One can readily generalize the theory of Section 2 to higher dimensions, and we shall do this here.Noncommutative Bianchi identities will also be obtained.

Again for the sake of explicitness we restrict attention to the Moyal product on the smooth functions.However, as we shall see in Section 5, it will be necessary to consider more general star-products inorder to discuss “general coordinate transformations” of noncommutative surfaces.

4.1. Noncommutativen-dimensional surfaces.We take a regionU in Rn for a fixedn, and write the

coordinate oft ∈U as(t1, t2, . . . ,tn). Let A denote the set of the smooth functions onU taking valuesin R[[h]]. Fix any constant skew symmetricn×n matrix θ. The Moyal product onA is defined by thefollowing generalization of equation (2.1):

f ∗g = limt′→t

exp

(

h∑i j

θi j ∂i∂′j

)

f (t)g(t ′)(4.1)

for any f ,g ∈ A. Such a multiplication is known to be associative. Sinceθ is a constant matrix, theLeibniz rule (2.3) remains valid in the present case:

∂i( f ∗g) = ∂i f ∗g+ f ∗ ∂ig.

For any fixed positive integerm, we can define a dot-product

• : Am⊗R[[h]] A

m−→Am(4.2)

by generalizing (2.4) toA•B = ai ∗bi for all A = (a1, . . . ,am) andB = (b1, . . . ,bm) in Am. As before,the dot-product is a map of two-sidedA-modules.

Assumem> n. ForX ∈Am, we letEi = ∂iX, and definegi j = Ei •E j . Denote byg = (gi j ) then×nmatrix with entriesgi j .

13

Definition 4.1. If g modh is invertible overU , we shall callX a noncommutative n-dimensional sur-faceembedded inAm, and callg the metric ofX.

The discussion on the metric in Section 2 carries through to the present situation, thus the invertibilityof g modh implies that there exists a unique inverse(gi j ). Now as in Section 2, we define the lefttangent bundleTX (respectively right tangent bundleTX) of the noncommutative surface as the left(respectively right)A-submodule ofAm generated by the elementsEi . The fact that the metricg belongsto GLn(A) enables us to show that the left and right tangent bundles areprojectiveA-modules.

The connection∇i on the left tangent bundle will be defined in the same way as in Section 2, namely,by the composition of the derivative∂i with the projection ofAm onto the left tangent bundle. Theconnection∇i on the right tangent bundle is defined similarly. Then∇i and∇i satisfy the analogousequation (5.6), and are compatible with the metric in the same sense as Proposition 2.7.

One can show that[∇i ,∇ j ] : TX−→ TX, [∇i , ∇ j ] : TX−→ TX

are left and rightA-module homomorphisms respectively. This allows us to define Riemann curvaturesof the tangent bundles as in equation (2.13). Then the formulae given in Lemma 2.12 are still valid whenthe indices in the formulae are assumed to take values in{1,2, . . . ,n}. Furthermore, the left and rightRiemann curvatures remain equal in the sense of Proposition2.13.

Remark4.2. One may define a dot-product• : Am⊗R[[h]] A

m−→Am with a Minkowski signature by

A•B= a0∗b0−m−1

∑i=1

ai ∗bi

for any A = (a0,a1, . . . ,am−1) andB = (b0,b1, . . . ,bm−1) in Am. This is still a map of two-sidedA-modules. Then the afore developed theory can be adapted to this setting, leading to a theory of anoncommutative surface embedded inAm with a Minkowski signature.

From the point of view of physics, noncommutative surfaces with Minkowski signature are moreinteresting. However, for the sake of being concrete, we shall consider only noncommutative surfaceswith Euclidean signature hereafter.

4.2. Bianchi identities. We examine properties of the Riemann curvature for arbitrary n andm. Themain result in this subsection are the noncommutative analogues of Bianchi identities.

DefineEi andEl as in (2.6). Then

∇pEl =−Γlpk∗Ek, ∇pEl =−Ek∗Γl

pk.(4.3)

These relations will be needed presently. Let

Rlki j ;p = ∂pRl

ki j −Γrpk∗Rl

ri j −Γrpi ∗Rl

r jk−Γrp j ∗Rl

rki +Rrki j ∗Γl

rp.(4.4)

Theorem 4.3. The Riemann curvature Rijkl satisfies the following relations

Rli jk +Rl

jki +Rlki j = 0, Rl

ki j ;p +Rlk jp;i +Rl

kpi; j = 0,(4.5)

which will be referred to as the first and second noncommutative Bianchi identities respectively.

Proof. It follows from the relation∇iE j = ∇ jEi that

[∇i ,∇ j ]Ek +[∇ j ,∇k]Ei +[∇k,∇i ]E j = 0.

This immediately leads to

g([∇i ,∇ j ]Ek, El )+g([∇ j ,∇k]Ei , E

l )+g([∇k,∇i ]E j , El ) = 0.

Using the definition of the Riemann curvature in this relation, we obtain the first Bianchi identity.

14

To prove the second Bianchi identity, note that

−∂pRlki j +g(∇p[∇i ,∇ j ]Ek, E

l )+g([∇i,∇ j ]Ek, ∇pEl ) = 0.

Cyclic permutations of the indicesp, i, j lead to two further relations. Adding all the three relationstogether, we arrive at

−∂pRlki j +g([∇i,∇ j ]∇pEk, E

l )+g([∇i,∇ j ]Ek, ∇pEl )

−∂iRlk jp +g([∇ j ,∇p]∇iEk, E

l )+g([∇ j ,∇p]Ek, ∇iEl )

−∂ jRlkpi +g([∇p,∇i ]∇ jEk, E

l )+g([∇p,∇i ]Ek, ∇ j El ) = 0,

(4.6)

where we have used the following variant of the Jacobian identity

∇p[∇i ,∇ j ]+ ∇i[∇ j ,∇p]+ ∇ j [∇p,∇i ] = [∇i ,∇ j ]∇p +[∇ j ,∇p]∇i +[∇p,∇i ]∇ j .

By a tedious calculation one can show that

Qi jkp := [∇ j ,∇k]∇pEi +[∇k,∇i ]∇pE j

+[∇p,∇k]∇iE j +[∇k,∇ j ]∇iEp

+[∇i ,∇k]∇ jEp+[∇k,∇p]∇ jEi

is identically zero. Now we addg(Qi jkp, El ) to the left-hand side of (4.6), obtaining an identity withfifteen terms on the left. Then the second Bianchi identity can be read off this equation by recalling(4.3). �

4.3. Einstein’s equation. Recall that in classical Riemannian geometry, the second Bianchi identitysuggests the correct form of Einstein’s equation. Let us make some preliminary analysis of this pointhere. As we lack guiding principles for constructing an analog of Einstein’s equation, the material ofthis subsection is of a rather speculative nature.

In Section 2, we introduced the Ricci curvatureRi j and scalar curvatureR. Their definitions can begeneralized to higher dimensions in an obvious way. Let

Rij = gik ∗Rk j,(4.7)

then the scalar curvature isR= Rii . Let us also introduce the following object:

Θlp := g([∇p,∇i ]E

i , El ) = gik ∗Rlkpi.(4.8)

In the commutative case,Θlp coincides withRl

p, but it is no longer true in the present setting. However,note that

Θll = gik ∗Rl

kli = gik ∗Rki = R.(4.9)

By first contracting the indicesj andl in the second Bianchi identity, then raising the indexk to i bymultiplying the resulting identity bygik from the left and summing overi, we obtain the identity

0 = ∂pR −∂iRip +g

(

[∇i ,∇l ]∇pEi , El)

+g(

[∇l ,∇p]∇iEi , El

)

−∂l Θlp +g

(

[∇i ,∇l ]Ei , ∇pEl

)

+g(

[∇p,∇i ]Ei , ∇l E

l)

+g(

[∇p,∇i ]∇l Ei , El

)

+g(

[∇l ,∇p]Ei , ∇iE

l)

.

Let us denote the sum of the last two terms on the right-hand side byϖp. Then

ϖp = gik ∗Rrkpl ∗Γl

ri − Γilr ∗grk ∗Rl

kpi.

15

In the commutative case,ϖp vanishes identically for allp. However in the noncommutative setting,there is no reason to expect this to happen. Let us now define

Rip;i = ∂iR

ip− Γi

pr ∗Rri + Γi

ir ∗Rrp,

Θlp;l = ∂l Θl

p−Θrl ∗Γl

rp + Θrp∗Γl

rl −ϖp.(4.10)

Then the second Bianchi identity implies

Rip;i + Θi

p;i− ∂pR= 0.(4.11)

The above discussions suggest that Einstein’s equation no longer takes its usual form in the noncom-mutative setting, but we have not been able to formulate a basic principle to enable us to write down afully consistent equation. However, formulae (4.11) and (4.9) seem to suggest that the following is areasonable noncommutative analogue of the Einstein equation in the vacuum:

Rij + Θi

j− δijR = 0.(4.12)

We were informed by J. Madore that in other contexts of noncommutative general relativity, it alsoappeared to be necessary to include an object analogous toΘi j in the Einstein equation.

5. GENERAL COORDINATE TRANSFORMATIONS

We investigate the effect of “general coordinate transformations” on noncommutativen-dimensionalsurfaces. This requires us to consider noncommutative surfaces defined overA endowed with star-products more general than the Moyal product. This should becompared with [8, 9, 2], where the only“general coordinate transformations” allowed were those keeping the Moyal product intact.

For the sake of being concrete, we assume that the noncommutative surface has Euclidean signature.

5.1. Gauge transformations. Denote byG(A) the set ofR[[h]]-linear mapsφ : A−→A satisfying thefollowing conditions

φ(1) = 1, φ = exp

(

∑i

εi∂i

)

modh,(5.1)

whereεi are smooth functions onU . Then clearly we have the following result.

Lemma 5.1. The setG(A) forms a subgroup of the automorphism group ofA asR[[h]]-module.

For any givenφ ∈ G(A), define anR[[h]]-linear map

∗φ : A⊗R[[h]] A−→A, f ⊗g 7→ f ∗φ g := φ−1 (φ( f )∗φ(g)) .(5.2)

Lemma 5.2. (1) The map∗φ is associative, thus there exists the associative algebra(A,∗φ) overR[[h]]. Furthermore,φ : (A,∗φ)−→ (A,∗) is an algebra isomorphism.

(2) Let∗φ = ∗φ−1, then for anyφ,ψ ∈ G(A)

ψ(

ψ−1( f )∗φ ψ−1(g))

= f ∗ψφ g.(5.3)

In this sense the definition of the new star-products respects the group structure ofG(A).

Proof. Because of the importance of this lemma for later discussions, we sketch a proof for it here, eventhough one can easily deduce a proof from [18].

For f ,g,h∈A, we have

( f ∗φ g)∗φ h = φ−1 ((φ( f )∗φ(g))∗φ(h))

= φ−1 (φ( f )∗ (φ(g)∗φ(h)))

= φ−1(φ( f )∗φ(g∗φ h))

= f ∗φ (g∗φ h),

16

which proves the associativity of the new star-product. Asφ is anR[[h]]-module isomorphism by def-inition, we only need to show that it preserves multiplications in order to establish the isomorphismbetween the algebras. Nowφ( f ∗φ g) = φ( f )∗φ(g). This proves part (1).

Part (2) can be proven by unraveling the left-hand side of (5.3). �

Adopting the terminology of Drinfeld from the context of quantum groups, we call an automorphismφ ∈ G(A) agauge transformation, and callG(A) thegauge group. The star product∗φ will be said to begauge equivalentto the Moyal product (4.1). However, note that our notion of gauge transformations isslightly more general than that in deformation theory [18],where the only type of gauge transformationsallowed in our context will be of the special form

φ = id + hφ1 + h2φ2 + . . . ,

whereφi areR-linear maps on the space of smooth functions onU such thatφi(1) = 0 for all i. Suchgauge transformations form a subgroup ofG(A).

Remark5.3. The prime aim of the deformation theory of [18] is to classifythe gauge equivalenceclasses of deformations in this restricted sense but for arbitrary associative algebras. The seminal paper[22] of Kontsevich provided an explicit formula for a star-product from each gauge equivalence class ofdeformations of the algebra of functions on a Poisson manifold.

Remark5.4. General star-products gauge equivalent to the Moyal product were evaluated explicitlyup to the third order inh in [29]. In [17], position-dependent star-products were also investigated andthe ultra-violet divergences of a quantumφ4 theory on 4-dimensional spaces with such products wereanalyzed.

Denoteui := φ−1(t i), i = 1,2, . . . ,n,

and refer tot 7→ u as ageneral coordinate transformationof U . DefineR[[h]]-linear operators onA by

∂φi = φ−1 ◦ ∂i ◦φ.(5.4)

Lemma 5.5. The operators∂φi have the following properties

∂φi ◦ ∂φ

j − ∂φj ◦ ∂φ

i = 0, ∂φi φ−1(t j) = δ j

i ,

and also satisfy the Leibniz rule

∂φi ( f ∗φ g) = ∂φ

i ( f )∗φ g+ f ∗φ ∂φi (g), ∀ f ,g∈A.

Proof. The proof is easy but very illuminating. We have

∂φi ◦ ∂φ

j − ∂φj ◦ ∂φ

i = φ−1◦ (∂i∂ j − ∂ j∂i)◦φ = 0.

Also, ∂φi φ−1(t j) = φ−1(∂it j) = δ j

i , sinceφ maps a constant function to itself.To prove the Leibniz rule, we note that

∂φi ( f ∗φ g) = φ−1 (∂i(φ( f )∗φ(g)))

= φ−1(∂iφ( f )∗φ(g))+ φ−1(φ( f )∗ ∂iφ(g))

= φ−1(

φ(∂φi f )∗φ(g)

)

+ φ−1(

φ( f )∗φ(∂φi g))

= ∂φi f ∗φ g+ f ∗φ ∂φ

i g.

This completes the proof of the lemma. �

The Leibniz rule plays a crucial role in constructing noncommutative surfaces over(A,∗φ).

17

5.2. Reparametrizations of noncommutative surfaces.The construction of noncommutative sur-faces over(A,∗) works equally well over(A,∗φ) for anyφ∈G(A). RegardAm⊗

R[[h]] Am as a two-sided

(A,∗φ)-module, and define the new dot-product

•φ : Am⊗R[[h]] A

m−→A

on it by A•φ B = ai ∗φ bi for anyA = (a1,a2, . . . ,am) andB = (b1,b2, . . . ,bm) in Am. It is obviously amap of two-side(A,∗φ)-modules. For an element

Xφ = (X1,X2, . . . ,Xm)

in the free two-sided(A,∗φ)-moduleAm, we define

Eφi = ∂φ

i Xφ, φgi j = Eφi •φ Eφ

j ,

whereφ acts onAm in a componentwise way. As in Section 2, letφg =

(φgi j)

i, j=1,...,n .

We shall say thatX is ann dimensional noncommutative surface with metricφg if φg modh is invertible.In this case,φg has an inverse

(φgi j)

.The left tangent bundleTXφ and right tangent bundleTXφ of Xφ are now respectively the left and

right (A,∗φ)-modules generated byEφi , i = 1,2, . . . ,n. The metricφg leads to a two-sided(A,∗φ)-module

mapφg : TXφ⊗

R[[h]] TXφ −→A, Z⊗W 7→ Z•φ W,

which is the restriction toTXφ⊗R[[h]] TXφ of the dot-product•φ. By using this map, we can decompose

Am into

Am = TXφ⊕ (TXφ)⊥, as left(A,∗φ)-module,

Am = TXφ⊕ (TXφ)⊥, as right(A,∗φ)-module,

where(TXφ)⊥ is orthogonal toTXφ and(TXφ)⊥ is orthogonal toTXφ with respect to the map inducedby the metric.

As in Definition 2.5, the operators

∇φi : TXφ −→ TXφ, ∇φ

i : TXφ −→ TXφ

are defined to be the compositions of∂φi with the projections ofAm onto the left and right tangent

bundles respectively. Thus, for anyZ ∈ TXφ and W ∈ TXφ,φg(∇φ

i Z,W) = ∂φi Z•φ W, φg(Z, ∇φ

i W) = Z•φ ∂φi W.(5.5)

By using the Leibniz rule for∂φi , we can show that the analogous equations of (2.8) are satisfied by

∇φi and∇φ

i , namely, for allZ ∈ TXφ, W ∈ TXφ and f ∈A,

∇φi ( f ∗φ Z) = ∂φ

i f ∗φ Z+ f ∗φ ∇φi Z,

∇φi (W ∗φ f ) = W∗φ ∂φ

i f + ∇φi W ∗φ f .

(5.6)

Furthermore, the operators are metric compatible:

∂φi

φg(Z, W) = φg(∇φi Z, W)+ φg(Z, ∇φ

i W), ∀Z ∈ TXφ, W ∈ TXφ.

Thus the two sets{∇φi } and{∇φ

i } define connections on the left and right tangent bundles respectively.The Christoffel symbolsφΓk

i j and φΓki j in the present context are also defined in the same way as

before:

∇φi Eφ

j = φΓki j ∗φ Eφ

k , ∇φi Eφ

j = Eφk ∗φ

φΓki j .

18

Then we haveφΓk

i j = ∂φi Eφ

j •φ Eφl ∗φ

φglk, φΓki j = φgkl ∗φ Eφ

l •φ ∂φi Eφ

j .

These formulae are of the same form as those in equation (2.10).By using (5.6), we can show that the maps[∇φ

i ,∇φj ] : TXφ −→ TXφ and[∇φ

i , ∇φj ] : TXφ −→ TXφ are

left and right(A,∗φ)-module homomorphisms respectively. Namely, for allZ ∈ TXφ, W ∈ TXφ andf ∈A,

[∇φi ,∇

φj ]( f ∗φ Z) = f ∗φ [∇φ

i ,∇φj ]Z,

[∇φi , ∇

φj ](W ∗φ f ) = [∇φ

i , ∇φj ]W ∗φ f .

Thus we can define the curvaturesφRli jk , φRl

i jk as before by

[∇φi ,∇

φj ]E

φk = φRl

ki j ∗φ Eφl ,

[∇φi , ∇

φj ]E

φk = Eφ

l ∗φφRl

ki j ,

and also construct their relatives such asφRi jkl , φRi jkl . ThenφRli jk , φRl

i jk are given by the formulae of

Lemma 2.12 with∗ replaced by∗φ, ∂i by ∂φi , Γ by φΓ andΓ by φΓ. Also, Proposition 2.13 is still valid

in the present case, andφRli jk , φRl

i jk satisfy the Bianchi identities (see Theorem 4.3).

Now we examine properties of noncommutative surfaces undergeneral coordinate transformations.Let X = (X1,X2, . . . ,Xm) be an element ofAm. We assume thatX gives rise to a noncommutativesurface over(A,∗). Then we have the following noncommutative surfaces

X =(φ(X1),φ(X2), . . . ,φ(Xm)), over (A,∗),Xφ =(X1,X2, . . . ,Xm), over (A,∗φ),

associated withX. We callXφ over(A,∗φ) thereparametrizationof the noncommutative surfaceX over(A,∗) in terms ofu = φ−1(t).

Remark5.6. Note thatAm is regarded as an(A,∗)-module when we study the noncommutative surfaceX, and regarded as an(A,∗φ)-module when we studyXφ. Thus even thoughX andXφ are the sameelement inAm, they have quite different meanings when the module structures ofAm are taken intoaccount.

Denote by ˆg the metric, byΓki j and ˆΓk

i j the Christoffel symbols, and byRli jk , ˆRl

i jk the Riemannian

curvatures ofX. They can be computed by usingn-dimensional generalizations of the relevant formulaederived in Section 2. The metric, curvature and other related objects of the noncommutative surfaceXφ

over(A,∗φ) are given in the last subsection.

Theorem 5.7. There exist the following relations:

φgi j = φ−1(gi j ),φgi j = φ−1(gi j ) ,

φΓki j = φ−1(Γk

i j ),φΓk

i j = φ−1( ˆΓki j ),

φRli jk = φ−1(Rl

i jk), φRli jk = φ−1( ˆRl

i jk).

Remark5.8. We shall see in the next section that this theorem leads to thestandard transformationrules for the metric, connection and curvature tensors in the commutative setting when we take the limith→ 0.

19

Proof of Theorem 5.7.Consider the first relation. Sinceφ−1 is an algebraic isomorphism from(A,∗) to(A,∗φ), we have

φ−1(gi j ) = φ−1(∂i X • ∂ j X) = φ−1(∂iX)•φ φ−1(∂ j X).

Usingφ−1(∂i X) = ∂φi X, we obtain

φ−1(Ei) = Eφi , ∀i.

Thusφ−1(gi j ) = Eφ

i •φ Eφj = φgi j .

Sinceφ maps 1 to itself, it follows thatφ−1(gi j ) = φgi j . Now

φ−1(Γki j ) = φ−1(∂i E j • El ∗ glk)

= φ−1(∂i E j)•φ φ−1(El )∗φ φ−1(glk)

= ∂φi Eφ

j •φ Eφl ∗φ

φglk

= φΓki j .

The other relations can also be proven similarly by using thefact thatφ−1 is an algebraic isomorphism.We omit the details. �

It is useful to observe how the covariant derivatives transform under general coordinate transforma-tions. We have

∇φi Eφ

j = ∂φi Eφ

j + φΓki j ∗φ Eφ

k

= φ−1(∂i E j)+ φ−1(Γki j ∗ Ek)

= φ−1(∇i E j),

where∇i is the covariant derivative in terms of the Christoffel symbols Γki j .

Remark5.9. The gauge transformationφ that procures the “general coordinate transformation” alsochanges the algebra(A,∗) to (A,∗φ), thus inducing a map between noncommutative surfaces definedover gauge equivalent noncommutative associative algebras. This is very different from what happensin the commutative case, but appears to be necessary in the noncommutative setting.

Remark5.10. Although the concept of covariance under the gauge transformationφ is transparent andthe considerations above show that our construction of noncommutative surfaces is indeed covariant un-der such transformations, it appears that the concept of invariance becomes more subtle. In the classicalcase, a scalar is a function on a manifold, which takes a valueat each point of the manifold. Invariancemeans that when we evaluate the function at ”the same point” on the manifold, we get the same value(a real or complex number) regardless of the coordinate system which we use for the calculation. In thenon-commutative case, elements ofA are not numbers. When a general coordinate transformation isperformed, the algebraic structure ofA changes. It becomes rather unclear how to compare elements intwo different algebras.

5.3. Comparison with classical case.One obvious question is the classical analogues of the differ-ential operators∂φ

i and the gauge transformations inG(A) which bring about the general coordinatetransformations. We address this question below. Morally,one should regardG(A) as the “diffeomor-phism group” of the surface, and∂φ

i as “differentiation” with respect to the new coordinateui := φ−1(t i).Consider an elementφ ∈ G(A). In the classical limit (that is,h = 0), φ obviously reduces to an

element exp(ν) in the diffeomorphism group ofU , wherev = εi(t)∂i is a smooth tangent vector field onU classically. Then for any two smooth functionsa(t) andb(t), the Leibniz rulev(ab) = v(a)b+av(b)

20

implies that exp(v)(ab) = (exp(v)a)(exp(v)b) at the classical level. By regarding a smooth functiona(t) as a power series int (or t translated by some constants) we then easily see that

exp(v)a(t) = a(exp(v)t) at the classical level.(5.7)

Now for all f (t) ∈A,

∂φi φ−1( f (t)) = φ−1(∂i f (t)) =

∂ f (u)

∂ui modh

=∂φ−1( f (t))

∂ui modh,

where we have used (5.7). Replacingf (t) by φ( f (t)) in the above computations we arrive at

∂φi f (t) =

∂ f (t)∂ui modh.

Using this result, we obtain

φgi j = φ−1(

φ(

∂X(t)∂ui

)

·φ(

∂X(t)∂u j

))

modh,

where the· on the right-hand side is the usual scalar product forRn. Up to h terms,

φ−1(

φ(

∂X(t)∂ui

)

·φ(

∂X(t)∂u j

))

=∂X(t)

∂ui ·∂X(t)∂u j modh,

henceφgi j =

∂t p

∂ui gpq(t)∂tq

∂u j modh.

This is the usual transformation rule for the metric if we ignore terms of order 1 or higher inh.It is fairly clear now that we shall also recover the usual transformation rules for the Christoffel

symbols and curvatures in the classical limith→ 0. We omit the proof.

6. NONCOMMUTATIVE SURFACES: SKETCH OF GENERAL THEORY

In the earlier sections, we presented a theory of noncommutative n-dimensional surfaces over a de-formation of the algebra of smooth functions on a regionU ⊂ R

n. This theory readily generalizes toarbitrary associative algebras with derivations. Below isa brief outline of the general theory.

Let A be an arbitrary unital associative algebra over a commutative ringk. We shall writeab as theproduct of any two elementsa,b ∈ A. Let Z(A) be the center ofA. Then the set of derivations ofA forms a leftZ(A)-module such that for any derivationd andz∈ Z(A), zd is the derivation whichmaps anya∈ A to zd(a). We assume thatA has a set∂i , i = 1,2, . . . ,n, of mutually commutative andZ(A)-linearly independent derivatives.

Let Am be the freeA-module of rankm. Define a dot product

• : Am⊗k Am−→A

byA•B= aibi for anyA= (a1, . . . ,am) andB= (b1, . . . ,bm) in Am. Fixedm> n. LetX = (X1,X2, . . . ,Xm)be an element ofAm for some . As before, we define

Ei = ∂iX = (∂iX1,∂iX

2, . . . ,∂iXm),(6.1)

and construct ann×n matrixg overA with entries

gi j = Ei •E j .(6.2)

We say thatX defines a noncommutative surface overA if g ∈ GLn(A), and call g the metric of thenoncommutative surface.

21

Clearly theZ(A)-linear independence requirement on the derivations is necessary in order for anyinvertibleg to exist. Ifg∈GLn(A), then

TX = {ziEi | zi ∈A}, TX = {Eiwi | zi ∈A}

are finitely generated projective (left or right)A-modules, which are taken to be the tangent bundles ofthe noncommutative surface. The metric defines a map

g : TX⊗k TX−→A, Z⊗W 7→ g(Z,W) = Z•W

of two-sideA-modules. We define connections

{∇i : TX−→ TX | i = 1, . . . ,n}, {∇i : TX−→ TX | i = 1, . . . ,n}on the left and right tangent bundles respectively by generalizing the standard procedure in the theoryof surfaces [15]:

∇i( f Z) = (∂i f )Z+ f ∇iZ, ∀ f ∈A,Z ∈ TX,

g(∇iE j ,El ) = ∂iE j •El ;(6.3)

and

∇i(W f) = W∂i f + ∇iW f, ∀ f ∈A,W ∈ TX,

g(E j , ∇iEl ) = E j • ∂iEl .(6.4)

Then the connections are compatible with the metric in the sense of Proposition 2.11.It can be shown that[∇i ,∇ j ] : TX −→ TX and [∇i , ∇ j ] : TX −→ TX are left and rightA-module

homomorphisms respectively. Thus one can define curvaturesof the connections on the left and righttangent bundles in the same way as in Sections 2 and 4 (see equation (2.13)). We shall not present thedetails here, but merely point out that the various curvatures still satisfy Propositions 2.13 and 4.3.

7. DISCUSSION AND CONCLUSIONS

Riemannian geometry is the underlying structure of Einstein’s theory of general relativity, and histor-ically the realization of this fact led to important furtherdevelopments. In this paper we have developeda Riemannian geometry of noncommutative surfaces as a first step towards the construction of a consis-tent noncommutative gravitational theory.

Our treatment starts from the simplest nontrivial examples, on which the general theory is gradu-ally elaborated. We begin by constructing a noncommutativeRiemannian geometry for noncommu-tative analogues of 2-dimensional surfaces embedded in 3-space, working over an associative algebraA, which is a deformation of the algebra of smooth functions ona region ofR2. On A3 we define a“dot-product” analogous to the usual scalar product for theEuclidean 3-space. An embeddingX of anoncommutative surface is defined to be an element ofA3 satisfying certain conditions. Partial deriva-tives ofX then generate a left and also a right projectiveA-module, which are taken to be the tangentbundles of the noncommutative surface. Now the dot-producton A3 induces a metric on the tangentbundles, and connections on the tangent bundles can also be introduced following the standard proce-dure in the theory of surfaces [15]. Much of the classical differential geometry for surfaces is shown togeneralize naturally to this noncommutative setting. We point out that the embeddings greatly help theunderstanding of the geometry of noncommutative surfaces.

From the noncommutative Riemannian geometry of the 2-dimensional surfaces we go straightfor-wardly to the generalization to noncommutative geometriescorresponding ton-dimensional surfacesembedded in spaces of higher dimensions. In higher dimensions, the Riemannian curvature becomesmuch more complicated, thus it is useful to know its symmetries. A result on this is the noncommutativeanalogues of Bianchi identities proved in Theorem 4.3.

22

We also observe that there exists another objectΘij (see (4.8)), which is distinct from the Ricci

curvatureRij but also reduces to the classical Ricci curvature in the commutative case. Contracting

indices in the second noncommutative Bianchi identity, we arrive at an equation involving “covariantderivatives” of bothRi

j andΘij . This appears to suggest that Einstein’s equation acquiresmodification

in the noncommutative setting, as shown in Subsection 4.3. Work along this line is in progress [7].A special emphasis is put on the covariance under general coordinate transformations, as the funda-

mental principle of general relativity. It is physically natural that under general coordinate transforma-tions, the frame-dependent Moyal star-product would change. In this spirit, we introduce in Section 5general coordinate transformations for noncommutative surfaces, in the form of gauge transformationson the underlying noncommutative associative algebraA, which change as well the multiplication ofthe underlying associative algebraA, turning it into another algebra nontrivially isomorphic to A. Bycomparison with classical Riemannian geometry, we show that the gauge transformations should beconsidered as noncommutative analogues of diffeomorphisms. We emphasize that in our constructionwe allow for all possible diffeomorphisms, and not only those preserving theθ-matrix constant, as hasbeen done so far in most of the literature in the field.

The results are eventually generalized to a theory of noncommutative Riemannian geometry ofn-dimensional surfaces over arbitrary unital associative algebras with derivations, briefly outlined in Sec-tion 6. Noncommutative surfaces should provide a useful test ground for generalizing Riemanniangeometry to the noncommutative setting. The ultimate aim isto obtain the noncommutative versionof gravitational theory, covariant under appropriately defined general coordinate transformations and,possibly, compatible with the gauging of the twisted Poincare symmetry, in analogy with the classicalworks of Utiyama [27] and Kibble [21].

Acknowledgement: We are much indebted to A. Chamseddine, J. Gracia-Bondıa and J. Madore for use-ful discussions and valuable suggestions on the manuscript. Partial financial support from the AustralianResearch Council, National Science Foundation of China (grants 10231050, 10421001), NKBRPC(2006CB805905), and Chinese Academy of Sciences is gratefully acknowledged.

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