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Some Notes on the Gh-Lifts of Affine Connections

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Some Notes on the Gh-Lifts of Affine Connections 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 symmet- ric 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 cotan- gent bundles have been studied by many authors [3,7,4,5]. In this paper, calculating the ∗ coefficients of gh−lift GH r of the affine connection via the musical isomorphism we have ∗ determined the gh−lift GH r of the affine connection r. After using the coefficients of the ∗ gh−lift GH r we have determined the torsion tensor and the curvature tensor of the gh−lift ∗ GH r of Levi-Civita connection. Finally we have investigated properties of the geodesic of ∗ the gh−lift GH r 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 = [x2M 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 2 TxM. The cotangent bundle on M is denoted by ∗ ∗ ∗ i i i i T M = [x2M 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 2 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 C1. We use the ranges of the index i being f1; :::; ng and the index i being fn + 1; :::; 2ng. 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 r of the affine connection r by transferring the horizontal lift of affine connection r 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 r be an affine connection on M manifold. The horizontal lift H r of the affine connec- tion r to the tangent bundle TM is formed with equation H r = C r − V R (3.1) R.C. Akpınar 35 where V R is vertical lift of curvature tensor of r and C r is complete lift of affine con- nection to the tangent bundle TM. And also the horizontal lift H r of the affine connection r provides the conditions H V H H rV X Y = 0; rV X Y = 0; (3.2) H V V H H H rH X Y = (rX Y ) ; rH X Y = (rX Y ) 1 for any X; Y 2 =0 (M). c h H C Let Γab be coefficient of r according to the local coordinates x on M. Let ΓAB be coefficient of H r according to the induced coordinates xh; yh to the tangent bundle H C H TM. The non-zero coefficients ΓAB of the horizontal lift r 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 r. ∗ The horizontal lift H r of the affine connection r to the cotangent bundle T ∗M is formed with equation ∗ ∗ ∗ H r = C r − V R (3.4) ∗ ∗ where r is symmetric. V R is vertical lift of curvature tensor of r and C r is complete lift of ∗ ∗ ∗ H C H r to the cotangent bundle T M. Let Γ AB be coefficient of r 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 r 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 r 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 r 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 rcgbk + Γckgbt + Γcbgtk − g rkgcb + Γkcgtb + Γkbgct a ak ak t ak ak t = −Γcb + g rcgbk + g Γcbgtk − g rkgcb − 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.
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