Some Notes on the Gh-Lifts of Affine Connections

Home , Musical isomorphism

Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Mathematics, 40 (4), 33-39 (2020). http://dx.doi.org/10.29228/proc.41

Some notes on the gh-lifts of afﬁne connections

Rabia Cakan Akpınar

Received: 05.11.2019 / Revised: 05.04.2020 / Accepted: 08.05.2020

Abstract. Let M be a differentiable manifold with dimension n and T ∗M be its cotangent bundle. In this paper, we determine the gh − lift of the afﬁne connection via the musical isomorphism on the cotangent bundle T ∗M. We obtain the torsion tensor, curvature tensor and geodesic curve of the gh − lift of Levi- Civita connection.

Keywords. Horizontal lift, connection, curvature tensor, musical isomorphism, geodesic.

Mathematics Subject Classiﬁcation (2010): 55R10, 53C05

1 Introduction

The connections were studied from an infinitesimal perspective in Riemannian geometry. Several authors investigated the lifts of connections on the different type bundle. Pogoda [8] defined the horizontal lift of basic connections of order r to a transverse naturel vector bundle and studied its properties. Salimov and Fattayev [9] determined the horizontal lift and complete lifts of the linear connection from a smooth manifold to its coframe bundle. Several authors investigated the lifts of connections on the cotangent bundle T ∗M. Yano and Patterson [10] applied the horizontal lifts which was defined by them to the symmetric connections on the cotangent bundle T ∗M. Kures [6] determined all natural operators transforming classical torsion-free linear connections on a manifold M into classical linear connections on the cotangent bundle T ∗M. Also the Riemannian manifolds and the cotangent bundles have been studied by many authors [3,7,4,5]. In this paper, calculating the ∗ coefficients of gh−lift GH ∇ of the affine connection via the musical isomorphism we have ∗ determined the gh−lift GH ∇ of the affine connection ∇. After using the coefficients of the ∗ gh−lift GH ∇ we have determined the torsion tensor and the curvature tensor of the gh−lift ∗ GH ∇ of Levi-Civita connection. Finally we have investigated properties of the geodesic of ∗ the gh−lift GH ∇ to the cotangent bundle T ∗M.

R.C. Akpınar Kafkas University, Faculty of Arts and Sciences, Department of Mathemetics, 36000, Kars, Turkey E-mail: [email protected] 34 Some notes on the gh-lifts of afﬁne connections

2 Preliminaries

Let M be a pseudo-Riemannian manifold with n dimension. The tangent bundle on M i i i i is denoted by TM = ∪x∈M TxM. The local coordinates on TM are (x , x ) = (x , y ) where xi are local coordinates on M and yi are vector space coordinates according i i ∂ to the basis ∂/∂x , i.e. yx = y ∂xi ∈ TxM. The cotangent bundle on M is denoted by ∗ ∗ ∗ i i i i T M = ∪x∈M Tx M. The local coordinates on T M are (x , xe ) = (x , pi) where x i are local coordinates on M and pi are vector space coordinates according to the basis dx , i ∗ r i.e. px = pidx ∈ Tx M. We denote by =s (M) the set of all tensor ﬁelds of type (r, s) on M. Throughout this paper we assume the manifolds, tensor ﬁelds and connections to be diferentiable of class C∞. We use the ranges of the index i being {1, ..., n} and the index i being {n + 1, ..., 2n}. Let g be a pseudo Riemannian metric. g] : T ∗M → TM is the musical isomorphism associated with g pseudo Riemannian metric with inverse given by g[ : TM → T ∗M. The musical isomorphism g] described by

] M m m m J j j j m j jm g :x ˜ = (x , x˜ ) = (x , pm) → x = (x , x ) = (δmx , y = g pm). (2.1) In music notation, the sharp symbol ] increase a note by a half step. And similar to this musical notation, the musical isomorphism g] increase the indice of the vector space coordinate. The musical isomorphism g[ is described by

[ I i i i i K k k¯ k i i g : x = (x , x ) = (x , y ) → x˜ = (x , x˜ ) = (δi x , pk = gkiy ). (2.2) In another music notation, the ﬂat symbol [ lowers a note by a half step. Similar to this musical notation, the musical isomorphism g[ low the indice of the vector space coordinate. The Jacobian matrice of g[ is obtained by

M m [ M ∂x˜ δj 0 (g∗) = AeJ = = s (2.3) ∂xJ y ∂jgms gmj and the Jacobian matrice of g] is obtained by

J j ] J ∂x δm 0 (g∗) = AM = M = js jm , (2.4) ∂x˜ ps∂mg g where δ is the Kronecker delta [1]. Yano and Patterson [10] defined the horizontal lift and applied this notion to connection ∗ in M. The gh−lift is newly defined in [2]. We obtain gh-lift GH ∇ of the affine connection ∇ by transferring the horizontal lift of affine connection ∇ from tangent bundle to cotangent bundle via the musical isomorphism. The problems of transferring the lifts to the cotangent bundle were studied in [1,2].

3 The gh−lift of afﬁne connection

Let ∇ be an afﬁne connection on M manifold. The horizontal lift H ∇ of the afﬁne connection ∇ to the tangent bundle TM is formed with equation

H ∇ = C ∇ − V R (3.1) R.C. Akpınar 35 where V R is vertical lift of curvature tensor of ∇ and C ∇ is complete lift of afﬁne connection to the tangent bundle TM. And also the horizontal lift H ∇ of the afﬁne connection ∇ provides the conditions

H V H H ∇V X Y = 0, ∇V X Y = 0, (3.2) H V V H H H ∇H X Y = (∇X Y ) , ∇H X Y = (∇X Y ) 1 for any X,Y ∈ =0 (M). c h H C Let Γab be coefficient of ∇ according to the local coordinates x on M. Let ΓAB be coefficient of H ∇ according to the induced coordinates xh, yh to the tangent bundle H C H TM. The non-zero coefficients ΓAB of the horizontal lift ∇ to the tangent bundle TM are given by

H Γ c = Γ c , H Γ c = Γ c , H Γ c = Γ c , H Γ c = ys∂ Γ c − ysRc ab ab ab ab ab ab ab s ab sab (3.3) c where Rsab is the curvature tensor of ∇. ∗ The horizontal lift H ∇ of the afﬁne connection ∇ to the cotangent bundle T ∗M is formed with equation

∗ ∗ ∗ H ∇ = C ∇ − V R (3.4) ∗ ∗ where ∇ is symmetric. V R is vertical lift of curvature tensor of ∇ and C ∇ is complete lift of ∗ ∗ ∗ H C H ∇ to the cotangent bundle T M. Let Γ AB be coefﬁcient of ∇ according to the induced ∗ h ∗ H C coordinates x , ph to the cotangent bundle T M. The non-zero coefﬁcients Γ AB of the ∗ horizontal lift H ∇ are given by

∗ ∗ ∗ ∗ H Γ c = Γ c , H Γ c = −Γ b , H Γ c = −Γ a , H Γ c = p −∂ Γ s + Γ s Γ k + Γ s Γ t ab ab ab ac ab cb ab s a bc kc ab bt ac (3.5) h ∗ according to the induced coordinates x , ph to the cotangent bundle T M [11]. 2 K ∂ xe We obtain the ∂xA∂xB that have components

∂2xk ∂2xk ∂2xk ∂2xk e = 0, e = 0, e = 0, e = 0 (3.6) ∂xa∂xb ∂xa∂xb ∂xa∂xb ∂xa∂xb 2 k 2 ks 2 k 2 k 2 k ∂ xe ps∂ g ∂ xe kb ∂ xe ka ∂ xe = , = ∂ag , = ∂bg , = 0. ∂xa∂xb ∂xa∂xb ∂xa∂xb ∂xa∂xb ∂xa∂xb ∗ ∗ GH C GH And using (2.3), (2.4) , (3.3) and (3.6), the coefﬁcients Γ AB of ∇ are obtained from equation

∗ ∂x˜M ∂x˜S ∂xC ∂xC ∂2 x˜K g[ H Γ C = ( GH Γ C ) = ( H Γ K + ). (3.7) ∗ AB AB ∂xA ∂xB ∂x˜K MS ∂x˜K ∂xA∂xB ∗ ∗ GH C GH The non-zero coefﬁcients Γ AB of ∇ are obtained as follows where A, B, ... = 1, ..., 2n: 36 Some notes on the gh-lifts of afﬁne connections

∗ GH c m s c k c Γ ab = δa δb δkΓms = Γab

∗ GH Γ c = δmgsbg Γ k + g ∂ gkb ab a ck ms ck a b kb = Γac − g ∂agck b bk bk = −Γac + g ∂cgak − g ∂kgac b bk t t bk t t = −Γac + g 5cgak + Γcagtk + Γckgat − g 5kgac + Γkagtc + Γkcgta b bk bk t bk bk t = −Γac + g 5c gak + g Γcagtk − g 5k gac − g Γkagtc b = −Γac

∗ GH c ma s k ka Γ ab = g δb gckΓms + gck∂bg a ka = Γcb − g ∂bgck a ak ak = −Γcb + g ∂cgbk − g ∂kgcb a ak t t ak t t = −Γcb + g ∇cgbk + Γckgbt + Γcbgtk − g ∇kgcb + Γkcgtb + Γkbgct a ak ak t ak ak t = −Γcb + g ∇cgbk + g Γcbgtk − g ∇kgcb − g Γkbgct a = −Γcb

∗ GH c m s t k m s t k t k mt s k Γ ab = δa δb y ∂kgtcΓms + δa δb gck y ∂tΓms − y Rtms + pt∂ag δb gckΓms m st k 2 tk +δa pt∂bg gckΓms + gckpt∂abg t k t k t k mt k st k = y (∂kgtc) Γab + gcky ∂tΓab − gcky Rtab + pt∂ag gckΓmb + pt∂bg gckΓas 2 tk +gckpt∂abg t k t k t k mt k st k = y (∂kgtc) Γab + gcky ∂tΓab − gcky Rtab + pt∂ag gckΓmb + pt∂bg gckΓas t t sk k ts k ts −pt ∂aΓbc − gckptΓbs∂ag − gckpt ∂aΓbs g − gckptΓbs∂ag t t k t k t k st k = −pt ∂aΓbc + y (∂kgtc) Γab + gcky ∂tΓab − gcky Rtab + pt ∂bg gckΓas t sk k ts −gckptΓbs∂cg − gckpt ∂aΓbs g t t k t k t k k ts = −pt ∂aΓbc + y (∂kgtc) Γab + gcky ∂tΓab − gcky Rtab − gckpt ∂aΓbs g st k t sk +pt ∂bg gckΓas − gckptΓbs∂ag t t k ts k ts k t k = −pt ∂aΓbc + y (∂kgtc) Γab + gck g ps∂tΓab − ptg ∂aΓbs − y Rtab s mt t sm k t s mk k ms +pt −Γbmg − Γbmg gckΓas − gckptΓbs −Γamg − Γamg t s k ts m k ts k k = −pt ∂aΓbc + psΓckΓab + g gmcpsΓkt Γab + gckg ps ∂tΓab − ∂aΓbt ts k t k ms s k tm −gckg psRtab − gckps ΓbmΓatg + ΓbmΓatg s t mk s k mt +ghkps ΓbtΓamg + ΓbtΓamg t s k s k s s s = −pt ∂aΓbc + psΓckΓab + psΓkcΓab + ps∂cΓab − ps∂aΓbc − ps∂cΓab s s m s m s t s m s t +ps∂aΓcb − psΓcmΓab + psΓamΓcb − psΓatΓbc − psΓbmΓac + psΓbtΓac s t +psΓbtΓac s s k s t = −ps∂aΓbc + psΓkcΓab + psΓbtΓac s s k s t = ps −∂aΓbc + ΓkcΓab + ΓbtΓac After these results we get the following theorem Theorem 3.1 Let (M, g) be a pseudo Riemannian manifold with dimension n. Let ∇ be an afﬁne connection on manifold M and H ∇ be the horizontal lift of ∇ to the tangent bundle ∗ TM. Then the differential of H ∇ by g[, i.e., a gh−lift GH ∇ to the cotangent bundle T ∗M ∗ coincides with the horizontal lift H ∇ to the cotangent bundle T ∗M if ∇ is a Levi-Civita connection. R.C. Akpınar 37

4 The Torsion and Curvature Tensors of gh− lift of Levi-Civita Connection

Let Te be torsion tensor of horizontal lift of the symmetric afﬁne connection ∇ on M to the ∗ cotangent bundle T ∗M . The torsion tensor Te of H ∇ is determined by

V V H V H H Te θ, ω = 0, Te X, ω = 0, Te X, Y = −γR (X,Y ) (4.1) H H 1 where R is the curvature tensor of ∇. X, Y are horizontal lift of X,Y ∈ =0 (M) V V 0 ∗ and θ, ω are vertical lift of θ, ω ∈ =1 (M) to the cotangent bundle T M. The non zero C component TeAB of Te is given

c t Teab = −ptRabc (4.2) h ∗ according to the induced coordinates x , ph to the cotangent bundle T M [11]. ∗ Let Tb be torsion tensor of the gh-lift GH ∇ of the Levi-Civita connection on M to the ∗ ∗ ∗ GH C GH cotangent bundle T M . Using the coefﬁcient Γ AB of ∇ to the cotangent bundle ∗ C T M the components Tb AB of Tb are obtained with the equation ∗ ∗ C GH C GH C Tb AB = Γ AB − Γ BA (4.3) h ∗ according to the induced coordinates x , ph to the cotangent bundle T M. We have

T c = 0, T c = 0, T c = 0, T c = −p Rt b ab b ab b ab b ab t abc (4.4) T c = 0, T c = 0, T c = 0, T c = 0. b ab b ab b ab b ab After we have ∗ Corollary 4.1 The torsion tensor Tb of the gh−lift GH ∇ of Levi-Civita connection coincides ∗ with the torsion tensor Te of horizontal lift H ∇ of the symmetric afﬁne connection.

Let Re be curvature tensor of horizontal lift of the symmetric afﬁne connection ∇ on M ∗ 1 0 to the cotangent bundle T M. For any X,Y,Z ∈ =0 (M) and ω, θ, ψ ∈ =1 (M), the ∗ curvature tensor Re of H ∇ provides the conditions

V V H V Re θ, ω = 0, Re X, ω = 0 (4.5) H H V V Re X, Y ψ = − (ψ ◦ R (X,Y )) H H H H Re X, Y Z = (R (X,Y ) Z)

C where R is the curvature tensor of ∇. The non zero components Re ABD of Re are given

c c Reabd = R abd (4.6) c s t s t Reabd = ps Γct R abd + Γdt R abc Rc = −R d eabd abc h ∗ according to the induced coordinates x , ph to the cotangent bundle T M. 38 Some notes on the gh-lifts of afﬁne connections

∗ Let Rb be curvature tensor of the gh-lift GH ∇ of Levi-Civita connection on M to the ∗ ∗ ∗ GH C GH cotangent bundle T M . Using the coefﬁcient Γ AB of ∇ to the cotangent bundle ∗ C T M the components Rb ABD of Rb are obtained with the equation

∗ ∗ ∗ ∗ ∗ ∗ C GH C GH C GH C GH T GH C GH T Rb ABD = ∂A Γ BD − ∂B Γ AD + Γ AT Γ BD − Γ BT Γ AD (4.7)

h according to the induced coordinates x , ph . We have

c c Rbabd = R abd (4.8) c s t s t Rbabd = ps Γct R abd + Γdt R abc Rc = −R d . babd abc After we have

∗ Corollary 4.2 The curvature tensor Rb of the gh−lift GH ∇ of Levi-Civita connection coin- ∗ cides with the curvature tensor Re of horizontal lift H ∇ of the symmetric afﬁne connection.

5 Geodesics of gh−lift of Levi-Civita Connection

∗ Let Ce be a geodesic curve of the gh-lift GH ∇ of Levi-Civita connection on M to the cotangent bundle T ∗M. The geodesic Ce is determined with the equations

d2xC ∗ dxA dxB + GH Γ C = 0 (5.1) dt2 AB dt dt c c c according to the induced coordinates x , x = (x , pc). ∗ Using the coefﬁcient of gh-lift GH ∇ we obtain following equations from (5.1):

c c c c 2 c ∗ a b ∗ a b ∗ a b ∗ b d x + GH Γ dx dx + GH Γ dx dx + GH Γ dx dx + GH Γ dxa dx = 0 dt2 ab dt dt ab dt dt ab dt dt ab dt dt (5.2) d2xc c dxa dxb dt2 + Γab dt dt = 0,

∗ c ∗ c ∗ c ∗ c d2xc GH dxa dxb GH dxa dxb GH dxa dxb GH dxa dxb dt2 + Γ ab dt dt + Γ ab dt dt + Γ ab dt dt + Γ ab dt dt = 0 2 a b b a d pc s s m s m dx dx a dpa dx b dx dpb dt2 + ps (−∂aΓbc + ΓcmΓab + ΓbmΓca ) dt dt − Γcb dt dt − Γac dt dt = 0 b b a d dpc a dx m dpm s dx dx dt dt − Γcbpa dt − Γac dt − Γbmps dt dt = 0 a (5.3) d δpc m δpm dx dt dt − Γac dt dt = 0 δ δpc dt dt = 0 2 δ pc dt2 = 0

b δpc dpc a dx where dt = dt − Γcbpa dt . After expressions (5.2) and (5.3) we have R.C. Akpınar 39

∗ Theorem 5.1 Let Ce be a geodesic according to the gh-lift GH ∇ of Levi-Civita connecion on M to the cotangent bundle T ∗M. The geodesic Ce has the eqations

d2xc c dxa dxb dt2 + Γab dt dt = 0, 2 δ pc dt2 = 0 c ∗ according to the induced coordinates (x , pc) to T M.

Theorem 5.2 The curve Ce on the cotangent bundle T ∗M is a geodesic according to the gh- ∗ lift GH ∇ of Levi-Civita connecion on M if the projection C = π Ce on M is a geodesic according to the ∇ on M and the second covariant differentiation of pc = pc (t) along C vanishes where π : T ∗M → M is the naturel projection.

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