Levi-Civita connections for conformally deformed metrics on tame differential calculi
aa r X i v : . [ m a t h . QA ] J a n Levi-Civita connections for conformally deformed metrics on tamedifferential calculi *Jyotishman Bhowmick, *Debashish Goswami and **Soumalya Joardar *Indian Statistical Institute, Kolkata, India** IISER Kolkata, Mohanpur, Nadia, India.Emails: [email protected], [email protected], [email protected]
Abstract
Given a tame differential calculus over a noncommutative algebra A and an A -bilinear pseudo-Riemannian metric g , consider the conformal deformation g = k.g , k being an invertible element of A . We prove that there exists a unique connection ∇ on the bimodule of one-forms of the differentialcalculus which is torsionless and compatible with g. We derive a concrete formula connecting ∇ and theLevi-Civita connection for the pseudo-Riemannian metric g . As an application, we compute the Ricciand scalar curvature for a general conformal perturbation of the canonical metric on the noncommutative2-torus as well as for a natural metric on the quantum Heisenberg manifold. For the latter, the scalarcurvature turns out to be a negative constant.
In recent years, the study of Levi-Civita connections in noncommutative geometry has attracted a lotof attention. Probes into the existence and uniqueness of Levi-Civita connections for noncommutativemanifolds go back to the works of [24], [15] and [25]. The recent surge in activities around similar questionswas pioneered by Rosenberg’s work ( [32] ) on the noncommutative torus where the existence of a uniqueLevi-Civita connection on a certain bimodule of derivations was proved. This line of attack was furtherpursued by Peterka and Sheu ( [29] ), Arnlind and his collaborators ( [1], [3] and references therein ) andmore recently by Landi and Arnlind ( [2] ).An alternative approach to the question of existence of Levi-Civita connections on the space of formswas taken by a number of mathematicians. In particular, bimodule Levi-Civita connections as well as ∗ -compatibility of Levi-Civita connections were studied by Beggs, Majid and their collaborators. We refer to[6], [27] and references therein and the book [9] for a comprehensive account. Following this line, existence ofa Levi-Civita connection on any quantum projective space ( for the Fubini-Study metric ) has been provedin [28]. For q -deformed connections on S q , we refer to the work of Landi, Arnlind and Ilwale ( [4] ) whilefor Levi-Civita connections on finite metric spaces and graphs, we refer to Chapter 8 of [9] and the paper[14] by Sitarz et al. Finally, yet another approach to study Levi-Civita connections on quasi-commutativealgebras has been initiated in [33] and [5].Tame differential calculi and metric-compatibility of connections on such calculi have been studied pre-viously in [10], [11] and [12]. As well-known by experts, ( see [9] ), if ∇ is a bimodule connection on thespace of one-forms E of any differential calculus, then ∇ induces a connection ∇ ( E⊗ A E ) ∗ on ( E ⊗ A E ) ∗ :=Hom A ( E ⊗ A E , A ) . In this article, we prove that this result is true for any connection provided the dif-ferential calculus is tame. Moreover, wre show that ∇ is compatible with a pseudo-Riemannian metric g ∈ Hom A ( E ⊗ A E , A ) in the sense of [11] and [10] if and only if ∇ ( E⊗ A E ) ∗ g = 0 . The goal of the rest ofthe article is to study Levi-Civita connections on conformal deformations of bilinear pseudo-Riemannianmetrics ( see Definition 5.1 ) on tame differential calculus. The main results of [10] and [11] were to provethe existence and uniqueness of Levi-Civita connections for any bilinear pseudo-Riemannian metric on tamedifferential calculi. In [13], this result was extended to the class of strongly σ -compatible pseudo-Riemannianmetric ( see Definition 3.1 of [13] ) which include bilinear pseudo-Riemannian metrics and their conformaldeformations. However, in this article, we present a very simple proof of the existence and uniqueness of Levi-Civita connections for such conformally deformed pseudo-Riemannian metric following a sufficient conditionestablished in [10]. The main benefit to our approach is that we can derive ( Theorem 5.4 ) a formula whichshows how the Levi-Civita connection deforms under the conformal deformation of the pseudo-Riemannianmetric. In case the bimodule of one-forms of the tame differential calculus under question is free, we can1lso derive the Christoffel symbols of the deformed Levi-Civita connection. Then we follow [11] to defineand study curvature of a Levi-Civita connection on tame differential calculi. We end the article with someconcrete examples of curvature computation.We should mention that in [11], a more direct and elegant proof of existence of Levi-Civita connectionsfor bilinear pseudo-Riemannian metrics was derived by imitating the classical proof and delivering a Koszul-formula for the connection. We have been unable to generalize the proof of [11] to conformally deformedmetrics which are typically not bilinear.Now we come to the class of examples for which our result is satisfied. We should admit that the classof tame differential calculi is restrictive. Nonetheless, we have several interesting examples for which werefer to Example 2.10 of [13]. This includes the differential calculi on the fuzzy 3 spheres and the quantumHeisenberg manifold coming from certain spectral triples defined in [24] and [15] respectively. We refer toTheorem 5.4 and Theorem 6.6 of [10] for the proofs. A differential calculus on the fuzzy 2-sphere wasalso proved to be tame in Theorem 8.5 of [11]. Theorem 3.4 of [26] proves that a differential calculusconstructed on the Cuntz algebra ( from a natural C ∗ -dynamical system ) is also tame.Another important class of examples comes from Connes-Landi deformations ( [18], [17] ) ofthe classical spectral triple of compact Riemannian manifolds equipped with a free and isometric toralaction. Indeed, if M is such a manifold, then the C ∗ -algebra C ( M ) can be deformed ( as in [31] ) to apossibly noncommutative C ∗ -algebra. The prescription of [18] delivers a canonical spectral triple over acanonical dense ∗ -subalgebra of this deformation. Theorem 7.1 of [10] verifies that the differential calculuscorresponding to this spectral triple is indeed tame.The plan of this article is as follows: in Section 2, we recall the definition and properties of tame differentialcalculi from [12]. In particular, we have pseudo-Riemannian metrics on such calculi. We begin Section 3by recalling from [11] ( and [10] ) the notion of metric-compatibility of a connection on a tame differentialcalculus. Then we show that if ∇ is a right connection on the bimodule of one-forms E of a tame differentialcalculus, then it induces a left connection ∇ ( E⊗ A E ) ∗ on the bimodule ( E ⊗ A E ) ∗ := Hom A ( E ⊗ A E , A ) . In [6]and [7], this was done for bimodule connections on any differential calculus. Thus, if g ∈ Hom A ( E ⊗ A E , A )is a pseudo-Riemannian metric on E , we can make sense of the equality ∇ ( E⊗ A E ) ∗ g = 0 and we demonstratethat ∇ ( E⊗ A E ) ∗ g = 0 if and only if ∇ is compatible with g in the sense of [11] ( and [10] ). In Section4 , we recall a sufficient condition for the existence of a Levi-Civita connection on any tame differentialcalculus. This criterion will be crucially used in the next section. In Section 5, we study Levi-Civitaconnections on conformal deformations of bilinear pseudo-Riemannian metrics. In the last section, wedefine and compute the Ricci and scalar curvatures for some examples including an arbitrary ‘conformaldeformation’ of the canonical metric on the noncommutative 2-torus as well as a bilinear metric on quantumHeisenberg manifolds. We begin by setting up the notations and conventions that are going to be followed. We will always workover the complex field. Unless otherwise mentioned, the symbol A will stand for a complex unital algebraand Z ( A ) will denote its center. The tensor product over the complex numbers C will be denoted by ⊗ C while the notation ⊗ A will denote the tensor product over the algebra A . If T is a linear map betweensuitable modules over A , Ran( T ) will denote the Range of T. The following well-known lemma will be useful for us.
Lemma 2.1 If E is a finitely generated projective right A -module, then there exist f , · · · f n in E and f , · · · f n in Hom A ( E , A ) such that for all f ∈ E and for all φ ∈ Hom A ( E , A ) ,f = X i f i f i ( f ) , φ = X i φ ( f i ) f i . { f i , f i : i = 1 , · · · n } is known as a pair of dual bases for E . Now we recall the definition of centered bimodules.2 efinition 2.2
We will say that a subset S of a right A -module E is right A -total in E if the right A -linearspan of S equals E . The center of an A -bimodule E is defined to be the set Z ( E ) = { e ∈ E : ea = ae ∀ a ∈ A} . It is easy to see that Z ( E ) is a Z ( A ) -bimodule. E is called centered if Z ( E ) is right A -total in E . If E and F are right A -modules, Hom A ( E , F ) will denote the set of all right A -linear maps from E to F . The set Hom A ( E , F ) has a natural A -bimodule structure which is as follows:the left A -module structure is given by left multiplication by elements of A , i.e, for elements a in A , e in E and T in Hom A ( E , F ) , ( a.T )( e ) := a.T ( e ) ∈ F . (1)The right A module structure on Hom A ( E , F ) is given by T.a ( e ) = T ( ae ) . (2)We will often make use of the following shorthand notation: Definition 2.3 If E is an A -bimodule, then E ∗ will stand for the A -bimodule Hom A ( E , A ) . The following isomorphism is well-known and will be used in the sequel:
Definition 2.4
Suppose E and F are finitely generated projective right A -modules. The map ζ E , F will denotethe canonical right A -module isomorphism from E ⊗ A F ∗ to Hom A ( F , E ) which is defined by the followingformula: ζ E , F ( X i e i ⊗ A φ i )( f ) = X i e i φ i ( f ) . Suppose A is an algebra. Then a differential calculus over A is a triplet (Ω ( A ) , ∧ , d ) where Ω( A ) is adirect sum of A -bimodules Ω j ( A ), with Ω ( A ) = A . The map ∧ : Ω( A ) ⊗ A Ω( A ) → Ω( A ) is an A -bimodulemap such that ∧ (Ω j ( A ) ⊗ A Ω k ( A )) ⊆ Ω j + k ( A ) .d is a map from Ω j ( A ) to Ω j +1 ( A ) such that d = 0 and d ( ω ∧ η ) = dω ∧ η + ( − deg( ω ) ω ∧ dη. Moreover, we will also assume that Ω j ( A ) is the right A -linear span of elements of the form da ∧ da ∧· · ·∧ da j . We will often denote the bimodule of one-forms Ω ( A ) of a generic differential calculus by the symbol E . We will always assume that E is finitely generated and projective as a right A -module. For notationalconvenience, we will sometimes denote a differential calculus by a pair ( E , d ) if E is the bimodule of one-formsof a differential calculus (Ω( A ) , ∧ , d ) . In Subsection 2.1, we define the notion of tame differential calculus and discuss some of its properties. InSubsection 2.2, we recall some results on pseudo-Riemannian metrics on tame differential calculi which willbe useful for us.
Definition 2.5
Suppose E is the bimodule of one-forms of a differential calculus (Ω( A ) , ∧ , d ) . We say thatthe differential calculus is tame if the following conditions hold:i. The bimodule E is finitely generated and projective as a right A module.ii. The following short exact sequence of right A -modules splits: → Ker( ∧ ) → E ⊗ A E → Ω ( A ) → . Thus, in particular, there exists a right A -module F isomorphic to Ω ( A ) such that: E ⊗ A E = Ker( ∧ ) ⊕ F (3)3 ii. The map u E : Z ( E ) ⊗ Z ( A ) A → E defined by u E ( X i e i ⊗ Z ( A ) a i ) = X i e i a i is an isomorphism of vector spaces,iv. Let us denote the idempotent in Hom A ( E ⊗ A E , E ⊗ A E ) with range Ker( ∧ ) and kernel F by the symbol P sym and define σ = 2 P sym − . We assume that σ satisfies the following equation for all ω, η ∈ Z ( E ) : σ ( ω ⊗ A η ) = η ⊗ A ω. (4)Let us note the following remark which will be very useful later on. Remark 2.6
Proposition 2.4 of [10] asserts that the map u E is actually a right A -linear isomorphism from Z ( E ) ⊗ Z ( A ) A to E . Moreover, in Lemma 4.4 of [10], it has been proved that the map P sym is A -bilinear.Thus, the same is true about the map σ = 2 P sym − . Examples of tame differential calculi have been discussed in the introduction. We refer to Example 2.10of [13] for more details. In Section 6, we will compute the scalar curvatures of the Levi-Civita connectionsfor some tame differential calculi on the noncommutative torus and the quantum Heisenberg manifold.Let us recall some properties of a tame differential calculus from [10], [11] and [13]. To begin with,Proposition 2.4 of [10] states that if E is the bimodule of one-forms of a tame differential calculus, then E iscentered ( see Definition 2.2 ). Thus, as proved in Section 4 of [11] ( see equation (2) ): a.e = e.a if e ∈ E and a ∈ Z ( A ) . (5)This implies that the bimodule of one-forms of a tame differential calculus is central in the sense of [21]and [22]. Moreover, all properties of centered bimodules naturally continue to hold for a tame differentialcalculus. In particular, we have the following lemma. Lemma 2.7 ( Lemma 1.4, [13] ) If E is the bimodule of one-forms of a tame differential calculus, then thefollowing statements hold:i. Z ( E ) is also left A -total in E . ii. The set { ω ⊗ A η : ω, η ∈ Z ( E ) } is both left and right A -total in E ⊗ A E . iii. If X is an element of E ⊗ A E , there exist v i in E , w i ∈ Z ( E ) and a i in A such that X = X i v i ⊗ A w i a i . iv. If in addition, E is a free right A -module with a basis { e , e , · · · , e n } ⊆ Z ( E ) , then any element X in E can be written as a unique linear combination P i,j e i ⊗ A e j a ij for some elements a ij in A . Now we explain the significance of the maps σ and P sym . The map σ plays the role of the flip map. In fact,for all ω ∈ Z ( E ) and e ∈ E , we have σ ( ω ⊗ A e ) = e ⊗ A ω, σ ( e ⊗ A ω ) = ω ⊗ A e (6)and hence P sym ( e ⊗ A ω ) = P sym ( ω ⊗ A e ) = 12 ( ω ⊗ A e + e ⊗ A ω ) (7)for all ω ∈ Z ( E ) and e ∈ E . The decomposition
E ⊗ A E = Ker( ∧ ) ⊕ F on simple tensors is explicitly given by ω ⊗ A ηa = 12 ( ω ⊗ A ηa + η ⊗ A ωa ) + 12 ( ω ⊗ A ηa − η ⊗ A ωa ) (8)for all ω, η in Z ( E ) and for all a in A . For the proof of these facts, we refer to Lemma 2.11 of [13].4 .2 Pseudo-Riemannian metrics on tame differential calculi
As discussed above, the map σ = 2 P sym − E ∗ := Hom A ( E , A )introduced in Definition 2.3. Definition 2.8 ( [10], [11] ) Suppose E is the bimodule of one-forms of a tame differential calculus (Ω( A ) , ∧ , d ) . A pseudo-Riemannian metric g on E is an element of Hom A ( E ⊗ A E , A ) such thati. g is symmetric, i.e. g ◦ σ = g, ii. g is non-degenerate, i.e, the right A -linear map V g : E → E ∗ defined by V g ( ω )( η ) = g ( ω ⊗ A η ) is anisomorphism of right A -modules.We will say that a pseudo-Riemannian metric g is a pseudo-Riemannian bilinear metric if in addition, g isalso left A -linear. Here, the right A -module structure on E ∗ = Hom A ( E , A ) is as in (2), i.e, if φ belongs to E ∗ , then for all a in A and e in E , ( φ.a )( e ) = φ ( a.e ) . We collect some results about pseudo-Riemannian metrics on tame differential calculus. These hadalready been proven in some form or the other in [11] and [12]. We also refer to Lemma 2.12 of [13] for acomplete proof.
Lemma 2.9
Suppose ( E , d ) is a tame differential calculus and g is a pseudo-Riemannian metric on E . Thenthe following are true:i. If either e or f belongs to Z ( E ) , we have g ( e ⊗ A f ) = g ( f ⊗ A e ) . (9) ii. If g is a pseudo-Riemannian bilinear metric, then g ( ω ⊗ A η ) ∈ Z ( A ) if ω, η belong to Z ( E ) . iii. If e is an element of E such that g ( e ⊗ A ω ) = 0 for all ω in Z ( E ) , then e = 0 . The same conclusionholds if g ( ω ⊗ A e ) = 0 for all ω in Z ( E ) . In this section, we study two definitions of metric-compatibility of connections on a tame differential calculus.In differential geometry, a connection ∇ : Ω ( M ) → Ω ( M ) ⊗ A Ω ( M ) canonically induces a connection ∇ (Ω ( M ) ⊗ C ∞ ( M ) Ω ( M )) ∗ on (Ω ( M ) ⊗ C ∞ ( M ) Ω ( M )) ∗ := Hom C ∞ ( M ) (Ω ( M ) ⊗ C ∞ ( M ) Ω ( M ) , C ∞ ( M )) . Thisis done by first extending ∇ to a connection on the bimodule Ω ( M ) ⊗ C ∞ ( M ) Ω ( M ) and then using thisconnection to define another connection on (Ω ( M ) ⊗ C ∞ ( M ) Ω ( M )) ∗ . Unfortunately, when A is a noncommutative algebra and ∇ is a connection on an A -bimodule F , thenthere is no canonical construction of a connection on F ⊗ A F . However, if ∇ is a bimodule connection, thenthis can be done. We refer to Proposition 2.3 of [6] for a proof. Consequently, Beggs and Majid defined thecompatibility condition of a connection ∇ on F with the metric g via the equality ∇ ( F⊗ A F ) ∗ g = 0 . We referto the paper [6] and also the book [9] for many interesting applications.We will show that if ( E , d ) is a tame differential calculus, the A -bimodule map σ : E ⊗ A E → E ⊗ A E introduced in Definition 2.5 helps us to circumvent the above mentioned problem. Thus, starting from an (ordinary ) connection on E , there is a recipe to define a connection on E ⊗ A E . This allows us to consider theequality ∇ ( F⊗ A F ) ∗ g = 0 . We then show ( Theorem 3.8 ) that this equality holds if and only if the connection ∇ on E is compatible with g in the sense of [11] ( and [10] ).We start with the definition of a connection and its torsion.5 efinition 3.1 ( [24], [16] ) Suppose (Ω( A ) , ∧ , d ) be a differential calculus on A and E := Ω ( A ) . A (right)connection on an A -bimodule F is a C -linear map ∇ : F → F ⊗ A E satisfying the equation ∇ ( f a ) = ∇ ( f ) a + f ⊗ A da for all f in F and a in A . The torsion of a connection ∇ on the bimodule E := Ω ( A ) is the right A -linear map T ∇ := ∧∇ + d :Ω ( A ) → Ω ( A ) . The connection ∇ is called torsionless if T ∇ = 0 . For us, a connection will always mean a right connection, unless otherwise mentioned. Let us recall that atame differential calculus always admits a torsionless connection on the bimodule of one-forms.
Theorem 3.2 ( Theorem 3.3 of [11] ) Suppose (Ω( A ) , ∧ , d ) is a tame differential calculus. Then the bimoduleof one-forms E = Ω ( A ) admits a torsionless connection. In what follows, for an A -bimodule F , the set F ∗ := Hom A ( F , A ) will be equipped with the A -bimodulestructure dictated by (1) and (2).We recall ( [7] ) that if ∇ is a ( right ) connection on a finitely generated and projective right A -module F , then it induces a left connection ∇ F ∗ on F ∗ , i.e, ∇ F ∗ is a C -linear map from F ∗ to E ⊗ A F ∗ such thatfor all φ in F ∗ and for all a in A , ∇ F ∗ ( aφ ) = a ∇ F ∗ ( φ ) + da ⊗ A φ. Definition 3.3 ( Subsection 3.2, [7] ) If ∇ : F → F ⊗ A E is a right connection on a finitely generatedand projective right A -module F with a pair of dual bases { f i , f i : i = 1 , , · · · n } as in Lemma 2.1 and ev : E ∗ ⊗ A E → A is the A -bilinear map defined by ev( φ ⊗ A e ) = φ ( e ) , then we define ∇ F ∗ : F ∗ → E ⊗ A F ∗ , ∇ F ∗ ( φ ) = X i [ d ( φ ( f i )) ⊗ A f i − (ev ⊗ A id ⊗ A id)( φ ⊗ A ∇ ( f i ) ⊗ A f i )] . The definition of ∇ F ∗ is independent of the choice of { f i , f i : i = 1 , , · · · n } as can be seen from the nextproposition. Proposition 3.4 ( [7] ) Suppose ( E , d ) is a any differential calculus and ∇ a connection on a finite generatedand projective right A -module F . If ζ E , F : E ⊗ A F ∗ → Hom A ( F , E ) is the isomorphism as in Definition 2.4and ev : E ∗ ⊗ A E → A the A -bilinear map as above, then for all φ ∈ F ∗ and all f ∈ E , we get ζ E , F ( ∇ F ∗ ( φ ))( f ) = d ( φ ( f )) − (ev ⊗ A id )( φ ⊗ A ∇ (f)) . (10) Thus, the definition of ∇ F ∗ is independent of the choice of the dual bases. Moreover, ∇ F ∗ is a left connectionon F ∗ . Proof:
This follows from the proof of Proposition 2.9 of [6]. However, for the sake of completeness, we givea proof. Indeed, for φ in F ∗ and f ∈ E , we make the following computation: ζ E , F ( ∇ F ∗ ( φ ))( f )= (id ⊗ A ev)( ∇ F ∗ ( φ ) ⊗ A f)= X i [ d ( φ ( f i )) f i ( f ) − (ev ⊗ A id)( φ ⊗ A ∇ (f i )f i (f))]= X i [ d ( φ ( f i ) f i ( f )) − φ ( f i ) d ( f i ( f )) − (ev ⊗ A id)( φ ⊗ A ∇ (f i f i (f))) + (ev ⊗ A id)( φ ⊗ A f i d(f i (f)))]= d ( φ ( f )) − X i φ ( f i ) d ( f i ( f )) − (ev ⊗ A id)( φ ⊗ A ∇ (f)) + X i φ (f i )d(f i (f))= d ( φ ( f )) − (ev ⊗ A id)( φ ⊗ A ∇ (f)) . ∇ F ∗ is a left connection on F ∗ : ∇ F ∗ ( aφ )= X i [ d ( aφ ( f i )) ⊗ A f i − (ev ⊗ A id ⊗ A id)( aφ ⊗ A ∇ ( f i ) ⊗ A f i )]= X i a [ d ( φ ( f i )) ⊗ A f i − (ev ⊗ A id ⊗ A id)( φ ⊗ A ∇ ( f i ) ⊗ A f i )] + X i daφ ( f i ) ⊗ A f i ( by the Leibniz rule and the left A − linearity of the map ev )= a ∇ F ∗ ( f ) + da ⊗ A φ by the second equation of Lemma 2.1. This completes the proof. ✷ Now we define the notion of the compatibility of a connection with a pseudo-Riemannian metric on atame differential calculus. We will need the following proposition:
Proposition 3.5 ( Subsection 4.1, [10] ) If g is a pseudo-Riemannian metric on the bimodule of one-forms E of a tame differential calculus ( E , d ) , we define Π g ( ∇ ) : Z ( E ) ⊗ C Z ( E ) → E as the map given by Π g ( ∇ )( ω ⊗ C η ) = ( g ⊗ A id) σ ( ∇ ( ω ) ⊗ A η + ∇ ( η ) ⊗ A ω ) . Then Π g extends to a well defined map from E ⊗ A E to E to be denoted by Π g ( ∇ ) . It turns out that for ω, η in Z ( E ) and a in A , the following equation holds:Π g ( ∇ )( ω ⊗ A ηa ) = Π g ( ∇ )( ω ⊗ Z ( A ) η ) a + g ( ω ⊗ A η ) da. (11)We recall that by Lemma 2.7, any element of E ⊗ A E is a finite sum of elements of the form ω ⊗ A ηa, where ω, η ∈ Z ( E ) and a belongs to A . Therefore, the equation (11) defines the map Π g ( ∇ ) on the whole of E ⊗ A E . Now we are in a position to define the metric-compatibility of a connection in our set-up.
Definition 3.6
Suppose ( E , d ) is a tame differential calculus and g is a pseudo-Riemannian metric on E . A connection ∇ on E is said to be compatible with g if Π g ( ∇ )( e ⊗ A f ) = d ( g ( e ⊗ A f )) for all e, f in E . A connection ∇ on E which is torsionless and compatible with g is called a Levi-Civita connection for thetriplet ( E , d, g ) . Let us introduce a Sweedler-type notation. If ∇ is a connection on E and e belongs to E , then we willwrite ∇ ( e ) = e (0) ⊗ A e (1) . (12)So if ω, η ∈ Z ( E ) and a ∈ A , thenΠ g ( ∇ )( ω ⊗ A ηa ) = g ( ω (0) ⊗ A η ) ω (1) a + g ( η (0) ⊗ A ω ) η (1) a + g ( ω ⊗ A η ) da. By ii. of Lemma 2.7, we know that any element of
E ⊗ A E is a finite linear combination of terms of the form ω ⊗ A ηa where ω, η ∈ Z ( E ) and a ∈ A . Hence, a connection ∇ is compatible with g if and only if d ( g ( ω ⊗ A ηa )) = g ( ω (0) ⊗ A η ) ω (1) a + g ( η (0) ⊗ A ω ) η (1) a + g ( ω ⊗ A η ) da (13)for all ω, η ∈ Z ( E ) and for all a ∈ E . Now we demonstrate that if ∇ is any ( right ) connection on the space of one-forms of a tame differentialcalculus ( E , d ) , then we can lift ∇ to a connection on E ⊗ A E . We do this in two steps: in the first step, wedefine a map e ∇ E⊗ A E : ( Z ( E ) ⊗ Z ( A ) Z ( E )) → ( E ⊗ A E ) ⊗ A E .
7n the second step, we define a map ∇ E⊗ A E : ( E ⊗ A E ) → ( E ⊗ A E ) ⊗ A E . Indeed, we define e ∇ E⊗ A E : Z ( E ) ⊗ Z ( A ) Z ( E ) → E ⊗ A E , e ∇ E⊗ A E ( ω ⊗ Z ( A ) η ) = σ ( ∇ ( ω ) ⊗ A η ) + ω ⊗ A ∇ ( η ) . We need to check that e ∇ E⊗ A E is well-defined. Thus, for ω, η in Z ( A ) and a in Z ( A ) , we compute ∇ E⊗ A E ( ωa ⊗ Z ( A ) η ) = σ ( ∇ ( ωa ) ⊗ A η ) + ωa ⊗ A ∇ ( η )= σ ( ∇ ( ω ) ⊗ A aη ) + ω ⊗ A η ⊗ A da + ω ⊗ A a ∇ ( η ) ( by (6) )= σ ( ∇ ( ω ) ⊗ A aη ) + ω ⊗ A η ⊗ A da + ω ⊗ A ∇ ( η ) a ( by (5) )= σ ( ∇ ( ω ) ⊗ A aη ) + ω ⊗ A ∇ ( ηa )= σ ( ∇ ( ω ) ⊗ A aη ) + ω ⊗ A ∇ ( aη ) ( as η ∈ Z ( E ) )= ∇ E⊗ A E ( ω ⊗ Z ( A ) aη )proving that e ∇ E⊗ A E is well-defined. Now we execute the second step. From Definition 4.7 of [10], we knowthat the map u E⊗ A E : Z ( E ) ⊗ Z ( A ) Z ( E ) ⊗ Z ( A ) A → E ⊗ A E , u E⊗ A E ( ω ⊗ Z ( A ) η ⊗ Z ( A ) a ) = ω ⊗ A ηa is an isomorphism. So it makes sense to define ∇ E⊗ A E : ( E ⊗ A E ) → ( E ⊗ A E ) ⊗ A E by the formula ∇ E⊗ A E ( u E⊗ A E ( ω ⊗ Z ( A ) η ⊗ Z ( A ) a )) = e ∇ E⊗ A E ( ω ⊗ Z ( A ) η ) a + ω ⊗ A η ⊗ A da. Proposition 3.7 If ( E , d ) is a tame differential calculus and ∇ a connection on E , then ∇ E⊗ A E is a well-defined connection on E ⊗ A E . Proof:
We start by proving that ∇ E⊗ A E is well-defined. From the defining formula of ∇ E⊗ A E , it is clearthat it suffices to prove ∇ E⊗ A E ( ω ⊗ Z ( A ) ηb ⊗ Z ( A ) a ) = ∇ E⊗ A E ( ω ⊗ Z ( A ) η ⊗ Z ( A ) ba ) (14)for all ω, η ∈ Z ( E ) , a ∈ A and for all b ∈ Z ( A ) . We compute ∇ E⊗ A E ( u E⊗ A E ( ω ⊗ Z ( A ) ηb ⊗ Z ( A ) a )) = e ∇ E⊗ A E ( ω ⊗ A ηb ) a + ω ⊗ A η ⊗ A bda = σ ( ∇ ( ω ) ⊗ A ηb ) a + ω ⊗ A ∇ ( ηb ) a + ω ⊗ A η ⊗ A bda = e ∇ E⊗ A E ( ω ⊗ Z ( A ) η ) ba + ω ⊗ A η ⊗ A d ( ba )( we have applied the Leibniz rules for ∇ and d )= ∇ E⊗ A E ( u E⊗ A E ( ω ⊗ Z ( A ) η ⊗ Z ( A ) ba )) . Now we prove that ∇ E⊗ A E is a connection. If ω, η ∈ Z ( E ) and a, b ∈ A , we obtain ∇ E⊗ A E (( ω ⊗ A ηa ) b ) = ∇ E⊗ A E u E⊗ A E ( ω ⊗ Z ( A ) η ⊗ Z ( A ) ab )= e ∇ E⊗ A E ( ω ⊗ Z ( A ) η ) ab + ω ⊗ A η ⊗ A d ( ab )= e ∇ E⊗ A E ( ω ⊗ Z ( A ) η ) ab + ω ⊗ A η ⊗ A dab + ω ⊗ A η ⊗ A adb = ∇ E⊗ A E ( ω ⊗ A ηa ) b + ( ω ⊗ A ηa ) ⊗ A db. This completes the proof. ✷ Summarizing, if ( E , d ) is a tame differential calculus and ∇ a ( right ) connection on E , then by Proposition3.7, we have a ( right ) connection ∇ E⊗ A E on E ⊗ A E . Then Proposition 3.4 delivers a left connection ∇ ( E⊗ A E ) ∗
8n (
E ⊗ A E ) ∗ . If g ∈ Hom A ( E ⊗ A E , A ) is a pseudo-Riemannian metric on E , we can therefore make sense ofthe quantity ∇ ( E⊗ A E ) ∗ . If { f i , f i : i = 1 , , · · · n } is a pair of dual bases for a finitely generated and projective right A -module F , then it can be easily checked that { f i ⊗ A f j , f i ⊗ A f j : i, j = 1 , , · · · n } is a pair of dual bases for F ⊗ A F . Therefore, Definition 3.3 implies that ∇ ( E⊗ A E ) ∗ g = X i,j [ d ( g ( f i ⊗ A f j )) ⊗ A ( f i ⊗ A f j ) − (ev ⊗ A id ⊗ A id)(g ⊗ A ∇ E⊗ A E (f i ⊗ A f j ) ⊗ A (f i ⊗ A f j ))] . Classically, it is well-known that a connection ∇ on the cotangent bundle Ω ( M ) of a manifold M iscompatible with a pseudo-Riemannian metric g if and only if ∇ (Ω ( M ) ⊗ C ∞ ( M ) Ω ( M )) ∗ g = 0 . We show thatthe same is true with our definition of metric-compatibility as in Definition 3.6.
Theorem 3.8 if ( E , d ) is a tame differential calculus and g is a pseudo-Riemannian metric on E , then aconnection ∇ on E is compatible with g in the sense of Definition 3.6 if and only if ∇ ( E⊗ A E ) ∗ g = 0 . Proof:
The proof follows from (10). Since { ω ⊗ A ηa : ω, η ∈ Z ( E ) , a ∈ A} is right A -total in E ⊗ A E ( partii. of Lemma 2.7 ) and ζ E , E⊗ A E : E ⊗ A ( E ⊗ A E ) ∗ → Hom A ( E ⊗ A E , E ) is an isomorphism ( Definition 2.4 ), ∇ ( E⊗ A E ) ∗ g = 0 if and only if ζ E , E⊗ A E ( ∇ ( E⊗ A E ) ∗ g )( ω ⊗ A ηa ) = 0for all ω, η ∈ Z ( E ) and for all a ∈ A . The equation (10) implies that ∇ ( E⊗ A E ) ∗ g = 0 if and only if for all ω, η and a as above, d ( g ( ω ⊗ A ηa )) = (ev ⊗ A id)(g ⊗ A ∇ E⊗ A E ( ω ⊗ A η a)) . But (ev ⊗ A id)(g ⊗ A ∇ E⊗ A E ( ω ⊗ A η a))= (ev ⊗ A id)(g ⊗ A ∇ E⊗ A E ( ω ⊗ A η )a + g ⊗ A ω ⊗ A η ⊗ A da)= (ev ⊗ A id)(g ⊗ A σ ( ∇ ( ω ) ⊗ A η )a + g ⊗ A ω ⊗ A ∇ ( η )a) + g( ω ⊗ A η )daby the definition of ∇ E⊗ A E . Using the Sweedler-type notation introduced in (12), the above expression isequal to g ( ω (0) ⊗ A η ) ω (1) a + g ( ω ⊗ A η (0) ) η (1) a + g ( ω ⊗ A η ) da = g ( ω (0) ⊗ A η ) ω (1) a + g ( η (0) ⊗ A ω ) η (1) a + g ( ω ⊗ A η ) da, since ω ∈ Z ( E ) and we have applied (9). Therefore, ∇ ( E⊗ A E ) ∗ g = 0 if and only if for all ω, η ∈ Z ( E ) and forall a ∈ A , d ( g ( ω ⊗ A ηa )) = g ( ω (0) ⊗ A η ) ω (1) a + g ( η (0) ⊗ A ω ) η (1) a + g ( ω ⊗ A η ) da. Comparing with (13), we deduce that ∇ is compatible with g if and only if ∇ ( E⊗ A E ) ∗ g = 0 . ✷ In [11] ( also see [10] ) and [13], existence and uniqueness of Levi-Civita connections have been proved for forbilinear and strongly σ -compatible pseudo-Riemannian metrics respectively. We will take the path adoptedin [10] and [13] to study Levi-Civita connections for conformally deformed pseudo-Riemannian metrics inSection 5. This will require a sufficient condition for the existence of Levi-Civita connections ( Theorem 4.2). We will need the following definition: Definition 4.1
For a tame differential calculus ( E , d ) , the symbol E ⊗ sym A E will denote Ker( ∧ ) = Ran( P sym ) . If g is a pseudo-Riemannian metric, the element dg will denote the map dg : E ⊗ A E → E , dg ( e ⊗ A f ) = d ( g ( e ⊗ A f )) . Theorem 4.2 ( Theorem 4.14, [10] ) Let ( E , d ) be a tame differential calculus and g a pseudo-Riemannianmetric on E . Let
E ⊗ sym A E be the A -bimodule of Definition 4.1. We define a map Φ g : Hom A ( E , E ⊗ sym A E ) → Hom A ( E ⊗ sym A E , E ) by the formula :Φ g ( L )( X ) = ( g ⊗ A id) σ ( L ⊗ A id)(1 + σ )( X ) for all X in E ⊗ sym A E . Then Φ g is right A -linear. Moreover, if Φ g : Hom A ( E , E ⊗ sym A E ) → Hom A ( E ⊗ sym A E , E ) is an isomorphismof right A -modules, then there exists a unique connection ∇ on E which is torsion-less and compatible with g. Moreover, if ∇ is a fixed torsionless connection on E , then ∇ is given by the following equation: ∇ = ∇ + Φ − g ( dg − Π g ( ∇ )) . (15) Here, dg : E ⊗ A E → E is the map defined in Definition 4.1.
The proof of this theorem works for any pseudo-Riemannian metric. The formula (15) follows from theproof of Theorem 4.14 of [10]. We only need to remark that the proof of Theorem 4.13 of [10] uses theexistence of a torsion-less connection on E . In our case, this condition is satisfied by virtue of Theorem 3.2.The main result of [11] and [10] is the following:
Theorem 4.3 ( Theorem 6.1 of [11], Theorem 4.1 of [10] ) Let ( E , d ) be a tame differential calculus and g be a pseudo-Riemannian bilinear metric on E . Then there exists a unique Levi-Civita connection for thetriplet ( E , d, g ) . Remark 4.4
In [10], Theorem 4.3 was proved by verifying that the map Φ g : Hom A ( E , E ⊗ sym A E ) → Hom A ( E ⊗ sym A E , E ) is an isomorphism of right A -modules. In [11], a completely different proof was given. Indeed, the uniqueness of such a connection followed byderiving a Koszul type formula of a torsionless and g compatible connection. The existence followed byproving that the above mentioned Koszul-formula indeed defines a torsionless and g -compatible connectionon E . We have been unable to generalize the proof of [11] for metrics which are not A -bilinear. However, wedemonstrate that for conformal deformations of a bilinear pseudo-Riemannian metrics on a tame differentialcalculus, Theorem 4.3 allows us to give a short proof for existence and uniqueness of Levi-Civita connection.This is the content of the next section. In [13], Theorem 4.3 was generalized to the case of strongly σ -compatible pseudo-Riemannian metrics. Let ( E , d ) be a tame differential calculus and g be a pseudo-Riemannian bilinear metric on E . We fix aninvertible element k of A . Definition 5.1
With ( E , d, g ) as above, the map g : E ⊗ A E → A , g ( e ⊗ A f ) = kg ( e ⊗ A f ) is called a conformal deformation of g . Indeed, it can be easily checked that g is a pseudo-Riemannian metric on E . Throughout this section, we will follow the notations developed till now so that g will denote a conformaldeformation of the pseudo-Riemannian bilinear metric g on a tame differential calculus ( E , d ) . Moreover,10he map dg : E ⊗ A E → E will be as defined in Definition 4.1. We note that if g = k.g is a conformaldeformation of g , then for all e, f in E ,dg ( e ⊗ A f ) = d ( g ( e ⊗ A f )) = d ( k.g ( e ⊗ A f )) = dk.g ( e ⊗ A f ) + k.dg ( e ⊗ A f )and so dg = dk.g + k.dg as maps from E ⊗ A E to E . (16)This section has three results. Firstly, in Theorem 5.2, we prove that there exists a unique Levi-Civitaconnections for the triplet ( E , d, g ) . Secondly, in Theorem 5.4, we derive a concrete formula for this Levi-Civita connection in terms of k and a fixed torsionless connection on E . Finally, Proposition 5.10 deducesthe Christoffel symbols for the Levi-Civita connection for the conformal deformation g when E is in additiona free right A -module satisfying some conditions.Let us clarify a couple of notations to be used in Theorem 5.2. We recall that E ⊗ sym A E is defined to beKer( ∧ ) . However, from the definition of a differential calculus, we know that the map ∧ : E ⊗ A E → Ω ( A )is an A -bimodule map and so E ⊗ sym A E = Ker( ∧ ) is an A -bimodule. Therefore, from (1), we know thatHom A ( E ⊗ sym A E , E ) is a left ( as well as a right ) A -module. For an invertible element k in A , let L k : Hom A ( E ⊗ sym A E , E ) → Hom A ( E ⊗ sym A E , E ) , L k ( X ) = k.X denote the left A -module multiplication on Hom A ( E ⊗ sym A E , E ) . For the pseudo-Riemannian bilinear metric g as above, dk.g : E ⊗ A E → E will denote the map defined by dk.g ( e ⊗ A f ) = dk.g ( e ⊗ A f ) . Since g is right A -linear, it is clear that dk.g is an element of Hom A ( E ⊗ A E , E ) . Since
E ⊗ sym A E = Ker( ∧ ) isboth a left and right A -submodule of E ⊗ A E by the above discussions, it makes sense to view the element dk.g as an element of Hom A ( E ⊗ sym A E , E ) . Thus, L k − ( dk.g ) is also an element of Hom A ( E ⊗ sym A E , E ) . Therefore,by Remark 4.4, we can conclude that Φ − g L k − ( dk.g ) is a well-defined element of Hom A ( E , E ⊗ sym A E ) . Wewill use these facts in Theorem 5.2.We have the following result:
Theorem 5.2
Let ( E , d ) be a tame differential calculus and g a pseudo-Riemannian bilinear metric on E . We will denote the Levi-Civita connection for the triplet ( E , d, g ) by the symbol ∇ g . If k is an invertibleelement of A , then there exists a unique Levi-Civita connection ∇ for the triplet ( E , d, k.g ) given by theformula: ∇ = ∇ g + Φ − g L k − ( dk.g ) . (17) Here, Φ g is the map defined in Theorem 4.2. Proof:
As stated above, g will denote the pseudo-Riemannian metric k.g . We use Theorem 4.2 to provethe existence and uniqueness of Levi-Civita connection for the triplet ( E , d, g ) . Thus, it suffices to prove thatthe map Φ g : Hom A ( E , E ⊗ sym A E ) → Hom A ( E ⊗ sym A E , E )is a right A -linear isomorphism. But since g = k.g , it is easy to verify that Φ g = L k . Φ g . By Remark 4.4,Φ g is a right A -linear isomorphism from Hom A ( E , E ⊗ sym A E ) to Hom A ( E ⊗ sym A E , E ) and so by Theorem 4.2,the Levi-Civita connection ∇ g exists.Since k is an invertible element of A , Φ g = L k Φ g is also invertible and its inverse is explicitly given by( L k . Φ g ) − = Φ − g ( L k ) − = Φ − g L k − . (18)In particular, the hypothesis of Theorem 4.2 is satisfied and we have a unique Levi-Civita connection for thetriplet ( E , d, g ) . Next, the equation (17) follows from (15). Indeed, the Levi-Civita connection ∇ g for the triplet ( E , d, g )is torsionless and so (18) implies that ∇ = ∇ g + Φ − g L k − ( dg − Π g ( ∇ g ) . (19)11ow since g = k.g , by using (11), it can be easily checked thatΠ g ( ∇ g ) = k. Π g ( ∇ g ) = k.dg since ∇ g is compatible with the metric g ( Definition 3.6 ). Hence, (19) implies that ∇ = ∇ g + Φ − g L k − ( dg − k.dg )= ∇ g + Φ − g L k − ( dk.g + k.dg − k.dg ) ( by (16) ) (20)= ∇ g + Φ − g L k − ( dk.g )This finishes the proof of the theorem. ✷ The formula (17) for the Levi-Civita connection ∇ in Theorem 5.2 can be made more explicit by derivinga formula for Φ − g . This is the content of Theorem 5.4 for which we need to need to make a definition.
Definition 5.3
For a pseudo-Riemannian bilinear metric g , we define an element Ω g ∈ E ⊗ A E by Ω g = (id E ⊗ A V − g ) ζ − E , E (id E ) . We recall that ζ E , E : Hom A ( E , E ) → E ⊗ A E ∗ is the the right A -module isomorphism defined in Definition2.4. Moreover, as g is bilinear , it can be easily checked that the map V g ( and hence V − g ) is left A -linear.Therefore, the map id E ⊗ A V − g makes sense.Now we are ready to state the following Theorem: Theorem 5.4
Suppose ( E , d ) is a tame differential calculus and g is a pseudo-Riemannian bilinear metricon E . If ∇ g is the Levi-Civita connection for ( E , d, g ) and k is an invertible element of A , then the Levi-Civita connection ∇ of ( E , d, kg ) is given by the following formula: ∇ ( ω ) = ∇ g ( ω ) + k − P sym ( dk ⊗ A ω ) − k − Ω g g ( dk ⊗ A ω ) . Here, the map P sym : E ⊗ A E → E ⊗ A E is the one defined in Definition 2.5. The proof of this theorem will be derived in steps in the next subsection. The Theorem 5.4 will help us toderive the Christoffel symbol of the Levi-Civita connection when the module is free. Φ g In this subsection, we prove Theorem 5.4. We will continue with the notations made before. In particular,we will be using the map P sym introduced in Definition 2.5 while Ω g is as defined in Definition 5.3.Comparing the statements of Theorem 5.2 and Theorem 5.4, it is clear that we need to prove the followingequation for all e in E :Φ − g ( L k − ( dk.g ))( e ) = k − P sym ( dk ⊗ A e ) − k − Ω g g ( dk ⊗ A e ) . Lemma 5.5 Ω g is an element of E ⊗ sym A E . Proof:
We will need the following A -bilinear map from [11] ( also see [10] ): g (2)0 : ( E ⊗ A E ) ⊗ A ( E ⊗ A E ) → A , g (2)0 (( ω ⊗ A η ) ⊗ A ( ω ′ ⊗ A η ′ )) = g ( ωg ( η ⊗ A ω ′ ) ⊗ A η ′ ) . (21)Then by Proposition 6.6 of [11] ( also see Proposition 3.8 of [10] ), we know that g (2)0 ( θ ⊗ A θ ′ ) = 0 ∀ θ ′ ∈ E ⊗ A E implies that θ = 0 . (22)12oreover, by Lemma 4.17 of [10], g (2)0 ( σ ( e ⊗ A f ) ⊗ A ( e ′ ⊗ A f ′ )) = g (2)0 (( e ⊗ A f ) ⊗ A σ ( e ′ ⊗ A f ′ )) (23)for all e, f, e ′ , f ′ belonging to E . Since
E ⊗ sym A E is the range of the idempotent P sym by definition, we need to prove that P sym (Ω g ) = Ω g . Since σ = 2 P sym − σ Ω g = Ω g . We claim that for all ω ′ , η ′ ∈ Z ( E ) ,g (2)0 ( σ Ω g ⊗ A ( ω ′ ⊗ A η ′ )) = g (2)0 (Ω g ⊗ A ( ω ′ ⊗ A η ′ )) . (24)If (24) is true, then by the right A -linearity of g (2)0 , we get that for all ω ′ , η ′ ∈ Z ( E ) and for all a in A ,g (2)0 (( σ Ω g − Ω g ) ⊗ A ( ω ′ ⊗ A η ′ a )) = 0 . By part ii. of Lemma 2.7, we deduce that g (2)0 (( σ Ω g − Ω g ) ⊗ A θ ′ ) = 0 for all θ ′ in E ⊗ A E . Therefore, (22) implies that σ Ω g = Ω g . Thus, we are left to prove (24).Once again we use Lemma 2.7 to recall that there exist v i in E , w i in Z ( E ) and a i in A such thatΩ g = X i v i ⊗ A w i a i . (25)On the other hand, by the definition of Ω g , for all e in E , we obtain e = ( ζ E , E (id E ⊗ A V g )Ω g )( e )= ( ζ E , E (id E ⊗ A V g )( X i v i ⊗ A w i a i ))( e )= X i ζ E , E ( v i ⊗ A V g ( w i a i ))( e )= X i v i V g ( w i a i )( e ) ( by Definition 2 . X i v i g ( w i a i ⊗ A e ) . (26)If ω ′ , η ′ ∈ Z ( E ) , then g (2)0 ( σ Ω g ⊗ A ( ω ′ ⊗ A η ′ )) = g (2)0 (Ω g ⊗ A σ ( ω ′ ⊗ A η ′ )) ( by (23) )= g (2)0 (Ω g ⊗ A ( η ′ ⊗ A ω ′ )) ( by (4) )= g (2)0 ( X i ( v i ⊗ A w i a i ) ⊗ A ( η ′ ⊗ A ω ′ )) ( by (25) )= g ( X i v i g ( w i a i ⊗ A η ′ ) ⊗ A ω ′ ) ( by (21) )= g ( η ′ ⊗ A ω ′ ) ( by (26) )= g ◦ σ ( η ′ ⊗ A ω ′ ) ( by Definition 2 . g ( ω ′ ⊗ A η ′ ) ( by (4) )= g ( X i v i g ( w i a i ⊗ A ω ′ ) ⊗ A η ′ ) ( by (26) )= g (2)0 (Ω g ⊗ A ( ω ′ ⊗ A η ′ )) ( by (21) ) . This finishes the proof of (24) and hence the lemma. ✷ emma 5.6 For all η in Z ( E ) , the following equation holds: ( g ⊗ A id) σ (Ω g ⊗ A η ) = η. Proof:
Let us continue writing Ω g as P i v i ⊗ A w i a i ( finitely many terms ) for some v i ∈ E , w i ∈ Z ( E ) and a i ∈ A as in Lemma 5.5 so that the relation σ Ω g = Ω g ( as obtained from Lemma 5.5 ) implies that X i v i ⊗ A w i a i = σ ( X i v i ⊗ A ω i a i ) = X i w i ⊗ A v i a i (27)as σ is right A -linear and we have applied the second equation of (6). Moreover, if η in Z ( E ) , we have( g ⊗ A id) σ (Ω g ⊗ A η ) = ( g ⊗ A id) σ ( X i v i ⊗ A w i a i ⊗ A η )( g ⊗ A id)( X i v i ⊗ A η ⊗ A w i a i ) ( by (6))= X i g ( v i ⊗ A η ) w i a i . (28)If η ′ belongs to Z ( E ) , we compute: g (( g ⊗ A id) σ (Ω g ⊗ A η ) ⊗ A η ′ ) = g ( X i g ( v i ⊗ A η ) w i a i ⊗ A η ′ ) ( by (28) )= g ( X i g ( v i ⊗ A η ) w i ⊗ A a i η ′ )= g ( X i w i g ( v i ⊗ A η ) ⊗ A a i η ′ ) ( as ω i ∈ Z ( E ) )= g (2)0 ( X i ( w i ⊗ A v i ) ⊗ A ( η ⊗ A a i η ′ )) ( by (21) )= g (2)0 ( X i ( w i ⊗ A v i ) a i ⊗ A ( η ⊗ A η ′ )) ( as η ∈ Z ( E ) )= g (2)0 ( X i ( v i ⊗ A w i a i ) ⊗ A ( η ⊗ A η ′ )) (by (27))= g ( X i v i g ( w i a i ⊗ A η ) ⊗ A η ′ )= g ( η ⊗ A η ′ ) (by (26)) . Therefore, for all η ′ in Z ( E ) , we obtain g ((( g ⊗ A id) σ (Ω g ⊗ A η ) − η ) ⊗ A η ′ ) = 0 . By part iii of Lemma 2.9, we can conclude that ( g ⊗ A id) σ (Ω g ⊗ A η ) = η. This proves the lemma. ✷ Having obtained the above results, we are now in a position to prove Theorem 5.4.
Proof of Theorem 5.4:
Let us define the map T dk : E ⊗ A E → E by T dk ( ω ⊗ A η ) = dkg ( ω ⊗ A η ) . By an abuse of notation, we will denote the restriction of T dk to E ⊗ sym A E by the same symbol T dk . We claimthat the following equation holds:(Φ − g ( T dk ))( ω ) = P sym ( dk ⊗ A ω ) −
12 Ω g g ( dk ⊗ A ω ) . (29)14f (29) holds, then the theorem follows from a computation. Indeed, we observe that the map Φ g is left A -linear since g is so and therefore,Φ − g ( k − dk.g )( ω )= k − Φ − g ( dk.g )( ω )= k − ( P sym ( dk ⊗ A ω ) −
12 Ω g g ( dk ⊗ A ω )) ( by (29) )so that by Theorem 5.2, ∇ ( ω ) = ∇ g ( ω ) + k − P sym ( dk ⊗ A ω ) − k − Ω g g ( dk ⊗ A ω ) . Thus, we are left to prove our claim, i.e, (29).We define L dk ∈ Hom A ( E , E ⊗ sym A E ) as L dk ( ω ) = P sym ( dk ⊗ A ω ) −
12 Ω g g ( dk ⊗ A ω ) . Since Ω g belongs to E ⊗ sym A E by Lemma 5.5, L dk ( ω ) indeed belongs to E ⊗ sym A E = Ran( P sym ) . We want to prove Φ g L dk = T dk . So for ω, η ∈ Z ( E ), we computeΦ g L dk ( ω ⊗ A η )= ( g ⊗ A id) σ ( L dk ⊗ A id)( ω ⊗ A η + η ⊗ A ω )= ( g ⊗ A id) σ ( L dk ( ω ) ⊗ A η + L dk ( η ) ⊗ A ω )= ( g ⊗ A id) σ ( P sym ( dk ⊗ A ω ) ⊗ A η −
12 Ω g g ( dk ⊗ A ω ) ⊗ A η + P sym ( dk ⊗ A η ) ⊗ A ω −
12 Ω g g ( dk ⊗ A η ) ⊗ A ω ) . Since ω, η belong to Z ( E ) , we can apply (7) to rewrite the above expression as:12 ( g ⊗ A id) σ ( dk ⊗ A ω ⊗ A η + ω ⊗ A dk ⊗ A η + dk ⊗ A η ⊗ A ω + η ⊗ A dk ⊗ A ω − Ω g g ( dk ⊗ A ω ) ⊗ A η − Ω g g ( dk ⊗ A η ) ⊗ A ω )= 12 [ g ( dk ⊗ A η ) ω + g ( ω ⊗ A η ) dk + g ( dk ⊗ A ω ) η + g ( η ⊗ A ω ) dk − ( g ⊗ A id) σ (Ω g ⊗ A η ) g ( dk ⊗ A ω ) − ( g ⊗ A id) σ (Ω g ⊗ A ω ) g ( dk ⊗ A η )]and in the last step we have used (6) as well as the fact that ω, η belong to Z ( E ) . Now using Lemma 5.6, (9) and the fact that ω, η ∈ Z ( E ), the expression reduces to 2 g ( ω ⊗ A η ) dk. However, since g is a bilinear metric and ω, η ∈ Z ( E ), by part ii. of Lemma 2.9 and (5), g ( η ⊗ A ω ) dk = dkg ( ω ⊗ A η ) . Hence for all ω, η ∈ Z ( E ), Φ g ( L dk )( ω ⊗ A η ) = T dk ( ω ⊗ A η ) . Since the set { ω ⊗ A η : ω, η ∈ Z ( E ) } is right A -total in E ⊗ A E ( part ii. of Lemma 2.7 ) and the mapsΦ g ( L dk ) , T dk are right A -linear, we can finally conclude that for all e, f in E , Φ g ( L dk )( e ⊗ A f ) = T dk ( e ⊗ A f ) . This proves our claim and hence the theorem. ✷ .2 Christoffel symbols for a class of free modules We end this section by Proposition 5.10 which computes the Christoffel symbols of the Levi-Civita connectionfor a diagonal metric on a class of free modules. The hypotheses of Proposition 5.10 ( and Proposition 5.8 )are satisfied for the differential calculi on the noncommutative torus and the quantum Heisenberg manifoldfor which we refer to Subsection 6.1 and Subsection 6.2. On the way, we will prove Proposition 5.8 whichestablishes a sufficient condition on the calculus ( E , d ) such that the Christoffel symbols are symmetric ( see(32) ). We start by recalling the definition of the Christoffel symbols of a connection. Definition 5.7
Suppose that E is a free module with a basis { e , e , · · · e n } ∈ Z ( E ) and ∇ is a connectionon E . Then we can define the “Christoffel symbols” Γ ijk ∈ A as follows: ∇ ( e i ) = X j,k e j ⊗ A e k Γ ijk . (30)We note that since ∇ ( e i ) belongs to E ⊗ A E , the elements Γ ijk are uniquely defined by part iv. of Lemma2.7. Proposition 5.8 If ( E , d ) is a tame differential calculus such that E is a free right A -module with a basis { e , e , · · · e n } ∈ Z ( E ) such that d ( e i ) = 0 for all i = 1 , , · · · n. Then we have the following:i. There exist derivations ∂ j : E → A , j = 1 , , · · · n, such that da = X j e j ∂ j ( a ) . (31) ii. The Christoffel symbols of a torsion-less connection satisfy Γ pkl = Γ plk for all p, k, l. (32) Proof:
Let a be in element of A . Hence, da belongs to E . Since ( E , d ) is a tame differential calculus, E iscentered ( see Subsection 2.1 ) and so there exist unique elements a i in A such that da = X j e j a j . For all j = 1 , , · · · n, we define ∂ j ( a ) := a j . We need to check that ∂ j is a derivation for all j. So we fix two elements a and b in A . Then by the definitionof ∂ j , we have X j e j ∂ j ( a.b ) = d ( a.b ) = da.b + a.db ( since d is a derivation )= X j e j ∂ j ( a ) .b + a X j e j ∂ j ( b ) = X j e j ∂ j ( a ) .b + X j e j a.∂ j ( b ) ( since e j belongs to Z ( E ) )= X j e j ( ∂ j ( a ) .b + a.∂ j ( b ))By comparing the coefficients of e j , we conclude that for all j = 1 , , · · · n,∂ j ( a.b ) = ∂ j ( a ) .b + a.∂ j ( b ) , i.e, ∂ j is a derivation. 16ow we prove the second assertion. We observe that by (7), P sym ( e i ⊗ A e j + e j ⊗ A e i ) = 12 ( e i ⊗ A e j + e j ⊗ A e i ) + 12 ( e j ⊗ A e i + e i ⊗ A e j ) = e i ⊗ A e j + e j ⊗ A e i . Therefore, e i ⊗ A e j + e j ⊗ A e i belongs to Ran( P sym ) = Ker( ∧ ) ( by Definition 2.5 ). Hence, e i ∧ e j = ∧ ( e i ⊗ A e j ) = − ∧ ( e j ⊗ A e i ) = − e j ∧ e i . (33)Since ∇ is assumed to be torsionless and d ( e i ) = 0 , we get0 = − d ( e i ) = ∧ ◦ ∇ ( e i )= ∧ ( X j,k e j ⊗ A e k Γ ijk ) = X j,k e j ∧ e k Γ ijk = X j The condition d ( e i ) = 0 is necessary for the equation (32) to hold. Indeed, consider thecomputation of the Levi-Civita connection for the fuzzy sphere in Section 8 of [11]. From Remark 8.7 of thatpaper, it is evident that the Christoffel symbols do not satisfy the relation (32) while equation ( 36 ) of [11]shows that d ( e m ) = 0 . This is a completely noncommutative phenomenon since in classical differential geometry, the equation (32) is always satisfied. Indeed, in the classical case, the Christoffel symbols are defined on a local chart ( U, x ) and the cotangent bundle is free over the open set U with a basis { e i := dx i : i = 1 , · · · , n } , n beingthe dimension of the manifold. Hence, de i = d x i = 0 . Proposition 5.10 Suppose ( E , d ) is a tame differential calculus and g is a bilinear pseudo-Riemannianmetric such that the following conditions are satisfied:1. E is a free right A -module with a basis { e , e , · · · e n } ∈ Z ( E ) such that d ( e i ) = 0 for all i = 1 , , · · · n. g ( e i ⊗ A e j ) = δ ij for all i, j. We will denote the Christoffel symbols of the Levi-Civita connection ∇ g for the triplet ( E , d, g ) by thesymbol (Γ ) ijl . Consider the conformally deformed metric g := k.g where k is an invertible element in A . Then the Christoffel symbols of the Levi-Civita connection ∇ for the triplet ( E , d, g ) are given by: Γ ijl = (Γ ) ijl + 12 ( δ il k − ∂ j ( k ) + δ ij k − ∂ l ( k ) − δ jl k − ∂ i ( k )) . (34) Here, ∂ i are the derivations as in (31) . roof: To begin with, we note that the existences of ∇ g and ∇ follow from Theorem 4.3 and Theorem5.4 respectively.We claim that under our assumptions, Ω g = X i e i ⊗ A e i . (35)Indeed, part iv. of Lemma 2.7 implies the existence of elements a ij in A such thatΩ g = X i,j e i ⊗ A e j a ij . Now for a fixed k , we have e k = ( g ⊗ A id) σ (Ω g ⊗ A e k ) ( by Lemma 5 . X i,j ( g ⊗ A id) σ ( e i ⊗ A e j a ij ⊗ A e k )= ( g ⊗ A id)( X i,j e i ⊗ A e k ⊗ A e j a ij ) ( by the second equation of (6))= X j e j a k j as g ( e i ⊗ A e k ) = δ i,k . Therefore, we can deduce that a k k = 1 and a k j = 0 if j = k . This proves theclaim.Now we apply Theorem 5.4 to see that ∇ ( e i ) = ∇ g ( e i ) + k − P sym ( dk ⊗ A e i ) − k − Ω g g ( dk ⊗ A e i )= ∇ g ( e i ) + 12 k − dk ⊗ A e i + 12 k − e i ⊗ A dk − k − Ω g g ( dk ⊗ A e i ) , (36)where, we have applied (7). Now, since e i belongs to Z ( E ) , k.e i = e i .k and so k − Ω g = k − ( X i e i ⊗ A e i ) ( by (35) )= ( X i e i ⊗ A e i ) k − = Ω g k − . Therefore, 12 k − dk ⊗ A e i + 12 k − e i ⊗ A dk − k − Ω g g ( dk ⊗ A e i )= 12 k − dk ⊗ A e i + 12 k − e i ⊗ A dk − 12 Ω g k − g ( dk ⊗ A e i )= 12 k − dk ⊗ A e i + 12 k − e i ⊗ A dk − X l e l ⊗ A e l g ( k − dk ⊗ A e i )( since g is left A − linear and we have applied (35) )= 12 ( X j k − e j ∂ j ( k ) ⊗ A e i + X j e i ⊗ A k − e j ∂ j ( k )) − 12 ( X l e l ⊗ A e l g ( k − ( X j e j ∂ j ( k )) ⊗ A e i ))( by (31) and as e i ∈ Z ( E ) )= X j,l e j ⊗ A e l ( 12 δ il k − ∂ j ( k ) + 12 δ ij k − ∂ l ( k ) − δ jl k − ∂ i ( k ))18s g ( e j ⊗ A e i ) = δ ij , e i ∈ Z ( E ) and g is A -bilinear.Hence, by (36), we obtain ∇ ( e i ) = X j,l e j ⊗ A e l (Γ ) ijl + X j,l e j ⊗ A e l ( 12 δ il k − ∂ j ( k ) + 12 δ ij k − ∂ l ( k ) − δ jl k − ∂ i ( k ))= X j,l e j ⊗ A e l [(Γ ) ijl + 12 ( δ il k − ∂ j ( k ) + δ ij k − ∂ l ( k ) − δ jl k − ∂ i ( k ))] . This completes the proof. ✷ We complete the section by proving that if ( E , d ) is a tame differential calculus such that E is a free right A -module admitting a central basis, then there indeed exists a bilinear pseudo-Riemannian metric g as inProposition 5.10. We are going to compute the curvature of the Levi-Civita connection of such a metricfor spectral triples on the noncommutative torus ( Subsection 6.1 ) and the quantum Heisenberg manifold( Subsection 6.2 ) using the above two results. We will also see that the condition d ( e i ) = 0 is satisfied forboth these examples. Lemma 5.11 Suppose ( E , d ) is a tame differential calculus such that E is a free right A -module with a basis { e , e , · · · e n } ∈ Z ( E ) . Then there exists a unique pseudo-Riemannian bilinear metric g on E such that g ( e i ⊗ A e j ) = δ ij . . Proof: We define g : E ⊗ A E → A , g (( X i e i a i ) ⊗ A ( X j e j b j )) = X i a i b i . (37)In particular, g ( e i ⊗ A e j ) = δ ij. It is clear that g is right A -linear.The uniqueness of the map g is clear as the facts that e i ∈ Z ( E ) , g is right A -linear and g ( e i ⊗ A e j ) = δ ij . g to be defined by (37). The fact that g is a pseudo-Riemannian metric has been proved inProposition 2.14 of [13]. So we only need to check that g is bilinear. As remarked above, g is right A -linearby definition. Let a be an element of A . Then g ( a ( X i e i a i ) ⊗ A ( X j e j b j )) = g (( X i e i aa i ) ⊗ A ( X j e j b j )) ( since e i belong to Z ( E ) )= X i aa i b i ( by (37) )= a ( X i a i b i )= ag (( X i e i a i ) ⊗ A ( X j e j b j ))which proves that g is bilinear. ✷ Following the groundbreaking work of Connes and Tretkoff in [20], computation of scalar curvature usingthe asymptotic expansion of the Laplace operator led to several seminal works. We refer to [19], [23] andreferences therein. In this section, we take an alternative path. We follow [11] to compute the curvature ofthe Levi Civita connection of a conformally deformed metric on a tame differential calculus. We will applyProposition 5.10 to compute the Ricci and scalar curvature for the module of one forms for the canonicalspectral triple on the noncommutative torus. The last subsection will deal with the computation of thecurvature for the space of one forms on the quantum Heisenberg manifold studied in [15]. Let us start bydefining the notions of Ricci and scalar curvature of a connection on a tame differential calculus ( E , d ) . Forthis, we need a few more definitions and some preparatory results.19irstly, let us recall ( Definition 2.5 ) that the short-exact sequence 0 → Ker( ∧ ) ι −→ E ⊗ A E ∧ −→ Ω ( A )splits, ι being the inclusion map. As a result, we have a direct sum decomposition E ⊗ A E = Ker( ∧ ) ⊕ F . Hence, ∧ : F → Ω ( A ) is a right A -linear isomorphism. Moreover, again by definition ( part iv. ofDefinition 2.5 ), P sym is an idempotent with range equals to Ker( ∧ ) and kernel equals to F . Hence, F =Ran(1 − P sym ) . Thus, we can view the map ∧ as a right A -linear isomorphism from Ran(1 − P sym ) to Ω ( A ) . In fact, to avoid confusion about the domain of the map ∧ , we will introduce the following notation. Definition 6.1 We will denote the restriction of the map ∧ to Ran(1 − P sym ) by the notation Q. We make the following observations about the map Q : Lemma 6.2 For a tame differential calculus ( E , d ) and Q as in Definition 6.1, we have the following:i. The map Q : Ran(1 − P sym ) → Ω ( A ) and its inverse Q − : Ω ( A ) → Ran(1 − P sym ) are A -bilinearmaps.ii. If X belongs to E ⊗ A E , then Q ((1 − P sym )( X )) = ∧ ( X ) . (38) iii. If a belongs to A and f belongs to E , then Q − ( da ∧ f ) = (1 − P sym )( da ⊗ A f ) . (39) Proof: By definition, the map Q is the restriction of the map ∧ to Ran(1 − P sym ) . But by the definition ofa differential calculus, the map ∧ : E ⊗ A E → Ω ( A ) is bilinear and so Q is bilinear. Consequently, Q − isalso bilinear.Next, if X belongs to E ⊗ A E , the equation (38) follows from the following computation: Q ((1 − P sym )( X )) = ∧ ((1 − P sym )( X )) = ∧ ( X ) − ∧ ( P sym ( X )) = ∧ ( X ) − P sym ( X ) belongs to Ran( P sym ) which is equal to Ker( ∧ ) by part iv. of Definition 2.5.Now we use (38) to prove (39). Since (1 − P sym )( X ) ∈ Ran(1 − P sym ) and Q : Ran(1 − P sym ) → Ω ( A )is a right A -linear isomorphism, (38) allows us to conclude that for all X ∈ E ⊗ A E ,Q − ( ∧ ( X )) = (1 − P sym )( X ) . (40)In particular, if a belongs to A and f belongs to E , then Q − ( da ∧ f ) = Q − ( ∧ ( da ⊗ A f )) = (1 − P sym )( da ⊗ A f ) . This completes the proof of the lemma. ✷ We will need another lemma to define the curvature operator. Lemma 6.3 Let ( E , d ) be a tame differential calculus and ∇ a connection on E . Then the map E ⊗ C E → E ⊗ A E ⊗ A E defined by e ⊗ C f (1 − P sym ) ( ∇ e ⊗ A f ) + e ⊗ A Q − ( df ) descends to a map from E ⊗ A E to E ⊗ A E ⊗ A E . We will denote this map by the symbol H. Moreover, if we define R ( ∇ ) := H ◦ ∇ : E → E ⊗ A E ⊗ A E , then R ( ∇ ) is a right A -linear map. roof: Let e, f ∈ E and a ∈ A . Then we get H ( e ⊗ C af ) = (1 − P sym ) ( ∇ ( e ) ⊗ A af ) + e ⊗ A Q − ( da ∧ f + a.df )= (1 − P sym ) ( ∇ ( e ) ⊗ A af ) + e ⊗ A (1 − P sym )( da ⊗ A f ) + e ⊗ A Q − ( adf )( by (39) )= (1 − P sym ) ( ∇ ( e ) a ⊗ A f ) + e ⊗ A (1 − P sym )( da ⊗ A f ) + ea ⊗ A Q − ( df )( since Q is left A − linear by Lemma . − P sym ) (( ∇ ( e ) a + e ⊗ A da ) ⊗ A f ) + ea ⊗ A Q − ( df )( as 1 − P sym is an idempotent )= (1 − P sym ) ( ∇ ( ea ) ⊗ A f ) + ea ⊗ A Q − ( df )= H ( ea ⊗ C f ) , which proves that H descends to a map from E ⊗ A E to E ⊗ A E ⊗ A E . Now we prove that the map R ( ∇ ) : E → E ⊗ A E ⊗ A E is a right A -linear map. Let e ∈ E and a ∈ A . Wewill use the Sweedler-type notation ∇ ( e ) = e (1) ⊗ A e (2) . It follows that R ( ∇ )( e ) = (1 − P sym ) ( ∇ ( e (1) ) ⊗ A e (2) ) + e (1) ⊗ A Q − ( d ( e (2) )) . (41)Therefore, using ∇ ( ea ) = ∇ ( e ) a + e ⊗ A da, it is easy to see that R ( ∇ )( ea ) = (1 − P sym ) ( ∇ ( e (1) ) ⊗ A e (2) a ) + e (1) ⊗ A Q − ( d ( e (2) a )) + (1 − P sym ) ( ∇ ( e ) ⊗ A da )= (1 − P sym ) ( ∇ ( e (1) ) ⊗ A e (2) ) a + e (1) ⊗ A [ Q − ( d ( e (2) ) a − e (2) ∧ da ) + (1 − P sym )( e (2) ⊗ A da )]( since P sym and Q − are right A − linear , see Lemma 6 . − P sym ) ( ∇ ( e (1) ) ⊗ A e (2) ) a + e (1) ⊗ A Q − ( d ( e (2) )) a ( by (40) )= R ( ∇ )( e ) a by (41), proving that R ( ∇ ) is right A -linear. This finishes the proof of the lemma. ✷ Now we are prepared to define the “curvature operator” following [11] and [15]. We observe that since R ( ∇ ) belongs to Hom A ( E , E ⊗ A E ⊗ A E ) by Lemma 6.3, we can apply the map ζ − E , E⊗ A E⊗ A E ( see Definition2.4 ) to R ( ∇ ) and the image lies in ( E ⊗ A E ⊗ A E ) ⊗ A E ∗ . Definition 6.4 If ( E , d ) is a tame differential calculus and ∇ is a torsionless connection on E , the curvatureoperator Θ of the connection ∇ is defined to be the image of the element R ( ∇ ) under the following maps: Hom A ( E , E ⊗ A E ⊗ A E ) ( E ⊗ A E ⊗ A E ) ⊗ A E ∗ E ⊗ A E ⊗ A E ⊗ A E ∗ . ζ − E , E⊗AE⊗AE σ Here, σ : E ⊗ A E ⊗ A E → E ⊗ A E ⊗ A E is the map id E ⊗ A σ. Now we proceed towards the definitions of the Ricci curvature and scalar curvature. We will need alemma whose proof is elementary: Lemma 6.5 If ( E , d ) is a tame differential calculus, u E : Z ( E ) ⊗ Z ( A ) A → E be the multiplication mapdefined in part iii. of Definition 2.5, v E : A ⊗ Z ( A ) Z ( E ) → E be defined by v E X i a i ⊗ Z ( A ) ω i ! = X i a i ω i andflip : Z ( E ) ⊗ Z ( A ) E ∗ → E ∗ ⊗ Z ( A ) Z ( E ) defined by flip ( e ′ ⊗ Z ( A ) φ ) = φ ⊗ Z ( A ) e ′ for all e ′ in Z ( E ) and φ in E ∗ , then the map ρ : E ⊗ A E ∗ → E ∗ ⊗ A E defined as the composition: E ⊗ A E ∗ Z ( E ) ⊗ Z ( A ) E ∗ E ∗ ⊗ Z ( A ) Z ( E ) = E ∗ ⊗ A ( A ⊗ Z ( A ) Z ( E )) E ∗ ⊗ A E . ( u E ) − ⊗ A id E∗ flip id E∗ ⊗ A v E is actually well-defined. roof: We only need to check that each of the maps ( u E ) − ⊗ A id E ∗ : E ⊗ A E ∗ → Z ( E ) ⊗ Z ( A ) E ∗ , id E ∗ ⊗ A v E : E ∗ ⊗ Z ( A ) Z ( E ) → E ∗ ⊗ A E and flip : Z ( E ) ⊗ Z ( A ) E ∗ → E ∗ ⊗ Z ( A ) Z ( E ) is well-defined.By Remark 2.6, we know that the map u E defined as u E ( X i ω i ⊗ Z ( A ) a i ) = X i ω i a i is a right A -linear isomorphism. Thus, the map ( u E ) − ⊗ A id E ∗ : E ⊗ A E ∗ → Z ( E ) ⊗ Z ( A ) E ∗ is well-defined.Moreover, an inspection of the proof of Proposition 2.4 of [10] shows that v E : A⊗ Z ( A ) Z ( E ) → E is a left A -linear, right Z ( A )-linear and invertible. Thus, the map id E ∗ ⊗ A v E : E ∗ ⊗ Z ( A ) Z ( E ) = E ∗ ⊗ A ( A⊗ Z ( A ) Z ( E )) →E ∗ ⊗ A E is well-defined.Finally, it can be easily checked that flip is well-defined ( and a right Z ( A )-linear isomorphism ). ✷ Now we are prepared to define the Ricci curvature and the scalar curvature. Definition 6.6 For a tame differential calculus ( E , d ) and a torsionless connection ∇ on E , the Ricci cur-vature Ric is defined as the element in E ⊗ A E given by Ric := (id E⊗ A E ⊗ A ev ◦ ρ )(Θ) , (42) where ev : E ∗ ⊗ A E → A is the A -bilinear map sending e ∗ ⊗ A f to e ∗ ( f ) for all e ∗ ∈ E ∗ and f ∈ E and Θ isthe curvature operator defined in Definition 6.4.Finally, the scalar curvature Scal is defined as: Scal := ev( V g ⊗ A id E )(Ric) ∈ A . (43) Remark 6.7 It is easy to see that in the classical case, i.e, when E = Ω ( A ) and A = C ∞ ( M ) , the abovedefinitions of Ricci and scalar curvature do coincide with the usual notions. Proposition 6.8 If ( E , d ) is a tame differential calculus such that E is a free right A -module with a basis { e , e , · · · e n } ∈ Z ( E ) such that d ( e i ) = 0 for all i = 1 , , · · · n. Then the curvature operator, Ricci tensorand the scalar curvature of a torsion-less connection ∇ are given by: R ( ∇ )( e i ) = X j,k,l e j ⊗ A e k ⊗ A e l r ijkl where r ijkl = 12 X p [(Γ pjk Γ ipl − Γ pjl Γ ipk ) − ∂ l (Γ ijk ) + ∂ k (Γ ijl )] . The Ricci tensor Ric is given by Ric = P j,l e j ⊗ A e l Ric( e j , e l ) where Ric( e j , e l ) = 12 X i [ X p (Γ pji Γ ipl − Γ pjl Γ ipi ) − ∂ l (Γ iji ) + ∂ i (Γ ijl )] . The Scalar curvature is given by Scal = P j,l g ( e j ⊗ A e l )Ric( e j , e l ) . Proof: We use the definition of H in Lemma 6.3 to compute R ( ∇ )( e i ) = H ◦ ∇ ( e i )= H ( X j,k e j ⊗ A e k Γ ijk )= X j,k [(1 − P sym ) ( ∇ ( e j ) ⊗ A e k Γ ijk ) + e j ⊗ A Q − ( d ( e k Γ ijk ))]= X j,k [ 1 − σ X m,n e m ⊗ A e n Γ jmn ⊗ A e k Γ ijk ) + e j ⊗ A Q − ( d ( e k )Γ ijk − e k ∧ d (Γ ijk ))]= X j,k,m,n − σ e m ⊗ A e n ⊗ A e k Γ jmn Γ ijk ) + X j,k e j ⊗ A Q − ( d ( e k ))Γ ijk − X j,k e j ⊗ A Q − ( e k ∧ d (Γ ijk )) (44)22ince e i belongs to Z ( E ) . Now, X n,k − σ e m ⊗ A e n ⊗ A e k Γ jmn Γ ijk )= 12 X n,k e m ⊗ A e n ⊗ A e k (Γ jmn Γ ijk − Γ jmk Γ ijn ) (45)by applying (6) as e n , e k belong to Z ( E ) . Next, as ∇ is a torsionless connection, we have Q − ( d ( e k )) = Q − ( − ∧ ◦∇ ( e k ))= − (1 − P sym ) ∇ ( e k ) ( by (40) )= − − σ X r,s e r ⊗ A e s Γ krs )= − X r,s e r ⊗ A e s (Γ krs − Γ ksr )by applying (6). However, Γ krs = Γ ksr by (32) and hence Q − ( d ( e k )) = 0 . (46)Finally, by applying (40), we get X k Q − ( e k ∧ d (Γ ijk ))= X k (1 − P sym )( e k ⊗ A d (Γ ijk )= 12 X k,n (1 − σ )( e k ⊗ A e n ∂ n (Γ ijk )) ( by (31) )= 12 X k,n e k ⊗ A e n ( ∂ n (Γ ijk ) − ∂ k (Γ ijn )) ( by (6) ) . (47)Plugging (45), (46) and (47) in (44), we obtain R ( ∇ )( e i )= 12 X j,k,m,n e m ⊗ A e n ⊗ A e k (Γ jmn Γ ijk − Γ jmk Γ ijn ) − X j,k,n e j ⊗ A e k ⊗ A e n ( ∂ n (Γ ijk ) − ∂ k (Γ ijn ))= X j,k,l e j ⊗ A e k ⊗ A e l [ 12 X p (Γ pjk Γ ipl − Γ pjl Γ ipk )] + X j,k,l e j ⊗ A e k ⊗ A e l . 12 [ − ∂ l (Γ ijk ) + ∂ k (Γ ijl )]= X j,k,l e j ⊗ A e k ⊗ A e l [ 12 X p (Γ pjk Γ ipl − Γ pjl Γ ipk ) − ∂ l (Γ ijk ) + ∂ k (Γ ijl ) ! ] . This completes the proof of the result. ✷ Remark 6.9 The only place where we have used the condition d ( e i ) = 0 is the equality Γ krs = Γ ksr whichproves that Q − ( d ( e k )) = 0 ( (46) ). Thus, in the absence of this condition, we would get some additionalterms. We recall that the noncommutative 2-torus C ( T θ ) is the universal C ∗ algebra generated by two unitaries U and V satisfying U V = e πiθ V U where θ is a number in [0 , . The ∗ - subalgebra A ( T θ ) of C ( T θ ) generatedby U and V will be denoted by A . 23e have the following concrete description of a spectral geometry on A : ( see [18] ):there are two derivations d and d on A obtained by extending linearly the rule: d ( U ) = U, d ( V ) = 0 , d ( U ) = 0 , d ( V ) = V. There is a faithful trace on A defined as follows: τ ( X m,n a mn U m V n ) = a , where the sum runs over a finite subset of Z × Z . Let H = L ( C ( T θ ) , τ ) ⊕ L ( C ( T θ ) , τ ) where L ( C ( T θ ) , τ ) denotes the GNS Hilbert space of A with respectto the state τ. We note that A is embedded as a subalgebra of B ( H ) by a (cid:18) a a (cid:19) . The Dirac operatoron H is defined by D = (cid:18) d + √− d d − √− d (cid:19) . Then, ( A , H , D ) is a spectral triple of compact type. In particular, for θ = 0 , this coincides with the classicalspectral triple on T . Let γ = (cid:18) (cid:19) and γ = (cid:18) √− −√− (cid:19) . The de-Rham differential d := d D : A → E := Ω ( A )is defined by d ( a ) = √− D, a ] . We have the following result: Proposition 6.10 The differential calculus ( E , d ) is tame. In fact, the bimodule E of one-forms is freelygenerated as a right A -module by the central elements e = 1 ⊗ C γ , e = 1 ⊗ C γ . The space of two forms is a rank one free module generated by e ∧ e . Moreover, we have d ( e ) = d ( e ) = 0 . (48) Proof: Consider the usual spectral triple on the 2-torus. Then the group T acts freely and isometricallyon T . The spectral triple on A defined above is the isospectral deformation ( [18] ) of the classical spectraltriple and hence we can apply Theorem 7.1 of [10] to conclude that ( E , d ) is tame. The structure of the spaceof one forms and two forms is well known and hence we omit the proof. It is also clear that e and e areelements of Z ( E ) . Thus, we are left with proving (48). Since γ .γ = − γ .γ and γ = 1 we have e ∧ e = 0 and e ∧ e = − e ∧ e . (49)Now, it is easy to see that dU = √− e U, dV = √− e V. Therefore, by Leibniz rule, we have0 = d U = d ( e ) .U − e ∧ dU = d ( e ) U − √− e ∧ e .U. By (49), we obtain d ( e ) U = 0 and hence d ( e ) = 0 since U is invertible. Similarly, d ( e ) = 0 . ✷ Remark 6.11 Proposition 6.10 allows us to apply Proposition 5.8 to the differential calculus ( E , d ) . Fromthe equalities dU = √− e U, dV = √− e V, it follows that the derivations ∂ and ∂ as in (31) are givenby the following formulas: ∂ ( U ) = √− U, ∂ ( V ) = 0 , ∂ ( U ) = 0 , ∂ ( V ) = √− V. From these formulas, it can be easily checked that ∂ and ∂ commute. heorem 6.12 Consider the differential calculus ( E , d ) on the noncommutative -torus A as above. Con-sider the pseudo-Riemannian bilinear metric g of Lemma 5.11. Let k be an invertible element of A . Thenthe Ricci and the scalar curvatures of the Levi-Civita connection for the triplet ( E , d, k.g ) are as follows: Ric( e , e ) = Ric( e , e ) = − 12 ( k − ( ∂ + ∂ )( k ) + ∂ ( k − ) ∂ ( k ) + ∂ ( k − ) ∂ ( k )) . Ric( e , e ) = − Ric( e , e ) = 12 ( ∂ ( k − ) ∂ ( k ) − ∂ ( k − ) ∂ ( k )) . Scal = − ( ∂ + ∂ )( k ) − k ( ∂ ( k − ) ∂ ( k ) − k∂ ( k − ) ∂ ( k )) . Proof: By virtue of Proposition 6.10, the hypotheses of Lemma 5.11 and Proposition 6.8 hold and so wecan apply them to this differential calculus. In particular, g ( e i ⊗ A e j ) = δ ij A (50)and Theorem 5.2 ensures the existence and uniqueness of the Levi-Civita connection for the triplet ( E , d, k.g )with Christoffel symbols as in (34).We will use Proposition 5.10 to compute the Christoffel symbols of the Levi-Civita connection for thetriplet ( E , d, g ) . Let ∇ g be the Levi-Civita connection for the triplet ( E , d, g ) as in Proposition 5.10. Weclaim that ∇ g ( e i ) = 0 for all i. (51)Indeed, we define a connection ∇ on E by the formula ∇ ( X i e i a i ) = X i e i ⊗ A da i . (52)In particular, ∇ ( e i ) = 0 for all i. We prove that ∇ is a torsionless and compatible with g so that the uniqueness of the Levi-Civitaconnection for a bilinear pseudo-Riemannian metric ( Theorem 4.3 ) will imply that ∇ = ∇ g . We have ∧ ◦ ∇ ( X i e i a i ) = X i ∧ ( e i ⊗ A da ) ( by (52) )= X i e i ∧ da i = − d ( X i e i a i )as d ( e i ) = 0 ( Proposition 6.10 ) proving that ∇ is torsionless.Next, by Definition 3.6, ∇ is compatible with g if for all e, f in E , Π g ( ∇ )( e ⊗ A f ) = d ( g ( e ⊗ A f )) . By (11), for all a in A , we haveΠ g ( ∇ )( X i,j e i ⊗ A e j a ) = X i,j Π g ( ∇ )( e i ⊗ A e j ) a + X i,j g ( e i ⊗ A e j ) da, where, from Proposition 3.5,Π g ( ∇ )( e i ⊗ A e j ) = ( g ⊗ A id) σ ( ∇ ( e i ) ⊗ A e j + ∇ ( e j ) ⊗ A e i ) = 025s ∇ ( e i ) = 0 for all i. Therefore,Π g ( ∇ )( X i,j e i ⊗ A e j a ) = 0 + X i,j g ( e i ⊗ A e j ) da = X i,j [ d ( g ( e i ⊗ A e j )) a + g ( e i ⊗ A e j ) da ] ( as g ( e i ⊗ A e j ) = δ ij . X i,j d ( g ( e i ⊗ A e j a )) ( as g is right A − linear and we have appplied Leibniz rule )= d ( g ( X i,j e i ⊗ A e j a )) . Hence, ∇ is compatible with g . By the discussion made above, ∇ = ∇ g and so (51) holds. By applying(34), we obtain Γ ijl = 12 ( δ il k − ∂ j ( k ) + δ ij k − ∂ l ( k ) − δ jl k − ∂ i ( k )) ,∂ i being the derivations as in (31). Thus, we have:Γ = 12 k − ∂ ( k ) , Γ = − k − ∂ ( k ) , Γ = Γ = 12 k − ∂ ( k ) , Γ = − k − ∂ ( k ) , Γ = 12 k − ∂ ( k ) , Γ = Γ = 12 k − ∂ ( k ) . Using these formulas for Christoffel symbols, we can compute ( using Proposition 6.8 ),Ric( e , e ) = X i,p =1 (Γ p i Γ ip − Γ p Γ ipi ) − X i =1 ( ∂ (Γ i i ) − ∂ i (Γ i ))= Γ Γ − Γ Γ + Γ Γ − Γ Γ − 12 ( ∂ ( k − ∂ ( k )) + ∂ ( k − ∂ ( k )))= 0 − ∂ ( k − ) ∂ ( k ) − k − ∂ ( k ) − ∂ ( k − ) ∂ ( k ) − k − ∂ ( k )= − 12 ( k − ( ∂ + ∂ )( k ) + ∂ ( k − ) ∂ ( k ) + ∂ ( k − ) ∂ ( k )) . The computations for Ric( e , e ) , Ric( e , e ) and Ric( e , e ) are similar and hence omitted. The only extraingredient in the computation of Ric( e , e ) and Ric( e , e ) is that the derivations ∂ and ∂ commute aswas remarked in Remark 6.11.Finally, using the formula of the scalar curvature in Proposition 6.8 and the equation (50), we get thatScal = X j,l kg ( e j ⊗ A e l )Ric( e j , e l )= X j k Ric( e j , e j )= − k ( k − ( ∂ + ∂ )( k ) + ∂ ( k − ) ∂ ( k ) + ∂ ( k − ) ∂ ( k ))= − ( ∂ + ∂ )( k ) − k ( ∂ ( k − ) ∂ ( k ) − k∂ ( k − ) ∂ ( k )) . ✷ In subsection 3.1 of [10], a canonical candidate g ′ for a pseudo-Riemannian bilinear metric on a tamespectral triple was proposed. It can be easily checked that for the spectral triple on the noncommutativetorus under consideration, g ′ is indeed a pseudo-Riemannian bilinear metric. The proof follows along thelines of Proposition 6.4 of [10]. It can be easily seen that g ′ ( e i ⊗ A e j ) = g ( e i ⊗ A e j ) = δ ij and so by theuniqueness of Lemma 5.11, g ′ = g . .2 Computation of the curvature for the example of the quantum Heisenbergmanifold In this subsection, we compute the curvature of the Levi Civita connection for a certain metric on the spaceof one-forms of the quantum Heisenberg manifold. The C ∗ -algebra of the quantum Heisenberg manifold wasdefined and studied in [30]. The differential calculus which we will consider comes from a spectral tripleconstructed in [15]. For the precise definition of the algebra of the quantum Heisenberg manifold and thespectral triple on it, we refer to [15]. The authors of [15] proved that there exists a pseudo-Riemannianmetric on the space of one forms for which there is no torsion-less connection which is also metric compatiblein the sense of [24]. However, using our definition of metric compatibility of a connection, it has been provedin [10] ( Theorem 6.4 ) that there exists a unique Levi-Civita connection for any pseudo-Riemannian bilinearmetric.In the rest of this subsection, we will be using the notations and results of Section 6 of [10] as well as[15]. In particular, we have the following: Proposition 6.13 ( [15], [10] ) Let ( E , d ) denote the differential calculus on the quantum Heisenberg man-ifold A as in [15]. Then the bimodule of one forms E is a free right A -module of rank . Moreover, E isgenerated by elements e , e , e belonging to Z ( E ) . The space of two forms Ω ( A ) is isomorphic to A ⊕ A ⊕ A . The differential calculus ( E , d ) is tame. The tameness of the differential calculus ( E , d ) is observed in the proof of Theorem 6.6 of [10].Let us fix a torsionless connection ∇ on ( E , d ) which we will need later. Since E := Ω ( A ) is a freemodule with generators e , e , e , any connection on E is determined by its action on e , e , e . Our choiceof the torsion-less connection ∇ is given by the following: ∇ ( e j ) = 0 for j = 1 , ∇ ( e ) = − e ⊗ A e . (53)The proof of the following proposition is a verbatim adaptation of the proof of Proposition 31 of [15]with the only difference that we use right connections instead of left connections. Proposition 6.14 ∇ is a torsion-less connection on E . Now we define a pseudo-Riemannian bilinear metric g on ( E , d ) . We will compute the Christoffel symbolsand scalar curvature of the Levi-Civita connection for the triplet ( E , d, g ) in Theorem 6.17. Lemma 6.15 Let g be the pseudo-Riemannian bilinear metric of Lemma 5.11 so that g ( e i ⊗ A e j ) = δ ij . If ∇ is the torsionless connection of Proposition 6.14 and Π g the map as in (11) , then we have the following: Π g ( ∇ )( e i ⊗ A e j ) = − X m e m T mij , where , T = T = 1 and T mij = 0 otherwise . Proof: Let us begin by remarking that it is easy to see ( from Proposition 6.13 ) that ( E , d ) satisfiesthe hypotheses of Lemma 5.11 so that the pseudo-Riemannian metric g makes sense. Secondly, sinceΠ g ( ∇ )( e i ⊗ A e j ) ∈ E and E is a free right A -module with basis e , e , e , the elements T mij exist uniquely.Since e , e , e ∈ Z ( E ) , we getΠ g ( ∇ )( e i ⊗ A e j ) = Π g ( ∇ )( e i ⊗ A e j ) = ( g ⊗ A id) σ ( ∇ ( e i ) ⊗ A e j + ∇ ( e j ) ⊗ A e i ) = Π g ( ∇ )( e j ⊗ A e i )by Proposition 3.5. Clearly , T mij = T mji . (54)From (53), it is immediate that for all i, j ∈ { , } , Π g ( ∇ )( e i ⊗ A e j ) = 0 . g ( ∇ )( e ⊗ A e ) = ( g ⊗ A id) σ ( ∇ ( e ) ⊗ A e + ∇ ( e ) ⊗ A e )= − ( g ⊗ A id) σ (( e ⊗ A e ) ⊗ A e )= − ( g ⊗ A id)( e ⊗ A e ⊗ A e ) ( by (6) )= − e . Thus, T = 1 , T = T = 0 . Next, Π g ( ∇ )( e ⊗ A e ) = ( g ⊗ A id) σ ( ∇ ( e ) ⊗ A e + ∇ ( e ) ⊗ A e )= − ( g ⊗ A id) σ (( e ⊗ A e ) ⊗ A e )= 0by (6) and as g ( e i ⊗ A e j ) = δ ij . . Thus, for all m = 1 , , , T m = 0 . The rest of the T mij can be computed by using (54). ✷ Remark 6.16 A candidate g ′ for a canonical pseudo-Riemannian bilinear metric on a tame spectral triplewas constructed in Subsection 3.1 of [10]. Proposition 6.4 of [10] verifies that g ′ satisfies all the requiredconditions to be a bilinear pseudo-Riemannian metric and moreover, the proof of this result shows that infact g ′ satisfies (37) . By the uniqueness of Lemma 5.11, it follows that g = g ′ . Now, we are ready to compute the explicit form of the Levi-Civita connection for the metric g on themodule E . Theorem 6.17 Consider the tame diferential calculus ( E , d ) on A as above, g be the pseudo-Riemannianbilinear metric of Lemma 6.15 and ∇ be the torsionless connection of Proposition 6.14. Then the uniqueLevi-Civita connection ∇ for the triplet ( E , d, g ) is given by ∇ = ∇ + L, where L : E → E ⊗ A E is defined by L ( e j ) = X i,m e i ⊗ A e m L jim , (55) L jim = 12 ( T mij + T ijm − T jmi ) , (56) where the elements { T mij : i, j, m = 1 , , } are as in Lemma 6.15.More precisely, the non zero L jim are as follows: L = L = 0 . , L = L = − . , L = L = 0 . , where we have denoted λ A simply by λ. If ∇ is given by ∇ ( e i ) = X j,k e j ⊗ A e k Γ ijk , (57) then the non zero Γ ijk are as follows: Γ = 1 , Γ = Γ = 0 . , Γ = 1 , Γ = − . , Γ = − . , Γ = 1 . , Γ = 0 . . roof: Theorem 6.6 of [10] proves that ( E , d ) is tame. By Remark 4.4, we know that the map Φ g defined inTheorem 4.2 is a right A -linear isomorphism. Therefore, by Theorem 4.2, there exists a unique Levi-Civitaconnection ∇ for the triplet ( E , d, g ) and is given by ∇ = ∇ + Φ − g ( dg − Π g ( ∇ )) . Let us define L = ∇ − ∇ . Then for all i, j and for all X in E ⊗ sym A E , Φ g ( ∇ − ∇ )( X ) = dg ( X ) − Π g ( ∇ )( X ) . (58)Morover, as ∇ and ∇ are both torsion-less connections, we get ∧ L ( e i ) = ∧∇ ( e i ) − ∧∇ ( e i ) = − d ( e i ) + d ( e i ) = 0 . (59)Let Q be the isomorphism from Ran(1 − P sym ) to Ω ( A ) as in Definition 6.1. Then0 = Q − ( ∧ ◦ L ( e j )) ( by (59) )= (1 − P sym ) L ( e j ) ( by (40) )= 1 − σ X i,m e i ⊗ A e m L jim ) ( by (55) )= 12 X i,m e i ⊗ A e m ( L jim − L jmi ) ( by (6) ) . Since { e i ⊗ A e j ; i, j } is a basis of E ⊗ A E , we obtain L jim = L jmi ∀ i, j, m. (60)Now we derive a relation between T mij and L ijm by the following computation: X m e m T mij = 12 X m e m T mij + 12 X m e m T mji ( by (54) )= − 12 Π g ( ∇ )( e i ⊗ A e j ) − 12 Π g ( ∇ )( e j ⊗ A e i ) ( Lemma 6 . 15 )= − Π g ( ∇ )( e i ⊗ A e j + e j ⊗ A e i d ( g ( e i ⊗ A e j + e j ⊗ A e i − Π g ( ∇ )( e i ⊗ A e j + e j ⊗ A e i g ( e i ⊗ A e j ) = δ ij . A )= Φ g ( L )( e i ⊗ A e j + e j ⊗ A e i e i ⊗ A e j + e j ⊗ A e i P sym ( e i ⊗ A e j ) ∈ Ran( P sym ) = E ⊗ sym A E )= ( g ⊗ A id) σ ( L ⊗ A id)(1 + σ )( e i ⊗ A e j + e j ⊗ A e i g in Theorem 4 . g ⊗ A id) σ ( L ⊗ A id)(1 + σ )( 1 + σ e i ⊗ A e j ) ( by (6) )= 2( g ⊗ A id) σ ( L ⊗ A id)( 1 + σ ( e i ⊗ A e j )= 2( g ⊗ A id) σ ( L ⊗ A id) P sym ( e i ⊗ A e j ) ( as P sym = 1 + σ g ⊗ A id) σ ( L ⊗ A id)( e i ⊗ A e j + e j ⊗ A e i )= ( g ⊗ A id) σ ( X k,x e k ⊗ A e x L ikx ⊗ A e j + X l,y e l ⊗ A e y L jly ⊗ A e i ) ( by (55) )29 ( g ⊗ A id)( X k,x e k ⊗ A e j ⊗ A e x L ikx + X l,y e l ⊗ A e i ⊗ A e y L jly ) ( by (6) )= X k δ jk X x e x L ikx + X l δ li X y e y L jly = X y e y ( L ijy + L jiy ) . This implies that L ij,m + L ji,m = T mij . (61)Interchanging ( i, j, m ) with ( j, m, i ) and ( m, i, j ), we have respectively: L jm,i + L mj,i = T ij,m , (62) L mi,j + L im,j = T jm,i . (63)Now, by (61) + (62) - (63) and (60), we have L ji,m = 12 ( T mij + T ijm − T jmi ) , which proves (56). The numerical expressions for L ij,m follow from the values of T mij in Lemma 6.15. Finally,since ∇ = ∇ + L, the Christoffel symbols Γ ijk as in (57) can be computed by using (53) and the values of L ij,m . ✷ Theorem 6.18 Let ∇ denote the Levi-Civita connection for the metric g on the module E of one formsover the quantum Heisenberg manifold A . The the Ricci and scalar curvature of ∇ are as follows: Ric( e , e ) = − , Ric( e , e ) = 1 , Ric( e , e ) = Ric( e , e ) = − . , Ric( e , e ) = − . , Ric( e , e ) = Ric( e , e ) = Ric( e , e ) = Ric( e , e ) = 0 . Scal = − . . 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