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The G-Lift of Affine Connection in the Cotangent Bundle

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The G-Lift of Affine Connection in the Cotangent Bundle Punjab University Journal of Mathematics ISSN: 1016-2526 Vol. 51(11)(2019) pp. 53-62 The g-lift of Affine Connection in The Cotangent Bundle Rabia Cakan and Esen Kemer Department of Mathemetics,Faculty of Arts and Sciences, 36000, Kafkas University,KARS, TURKEY, Email: [email protected] Email: [email protected] Received: 04 April, 2019 / Accepted: 27 June, 2019 / Published online: 01 October, 2019 Abstract. The g-lifts of affine connection and curvature tensor are ob- tained via the musical isomorphism in the cotangent bundle. AMS (MOS) Subject Classification Codes: 55R10; 53C15 Key Words: g-lift, complete lift, connection, curvature tensor, musical isomorphism, tan- gent bundle, cotangent bundle. 1. INTRODUCTION Musical isomorphism is provided between the tangent and cotangent bundles on the Riemannian Manifold. The exact origin of the term musical isomorphism is unknown. In 1971, the musical isomorphism was seen in the Berger and his collaborators’ study and the term musical isomorphism was not encountered before this year [3]. So it is thought that the musical isomorphism was first studied by Berger and his collaborators. The musical isomorphism defined by g metric tensor was extended by Poor [7]. Making use of the complete lifts on TM tangent bundles, Cakan and her collaborators construct the complete lifts of some tensor fields with different type on T ¤M cotangent bundles by means of a musical isomorphism [4]. The g-lifts of tensor fields are described via musical isomorphism newly by Cakan and Salimov [8]. Also the Riemannian manifolds and the tangent bundles studied a lot of authors [1, 2, 5, 6, 9, 10, 11, 12, 13] too. In the study that follows we solve ¤ two problems. Firstly we obtain the components of g¡lift of affine connection Gr via the musical isomorphism in the cotangent bundle T ¤M. Secondly we obtain components of ¤ g¡lift of curvature tensor GR via the musical isomorphism in the cotangent bundle T ¤M. 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 to the basis @=@xi, i.e. y = yi @ 2 T M. The cotangent bundle on M is denoted by x @xi x ¡ ¢ ¤ ¤ ¤ 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 , 53 54 Rabia Cakan and Esen Kemer i ¤ i.e. px = pidx 2 Tx M. 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] is described by ] M m m m J j j j m j jm g :x ~ = (x ; x~ ) = (x ; pm) ! x = (x ; x ) = (±mx ; y = g pm) where ± is the Kronecker delta. In music notation, the sharp symbol ] increases a note by a half step. Similar to this musical notation, the musical isomorphism g] increases the indice of the vector space co- ordinate. The musical isomorphism g[ is described by [ I i i i i K k k¹ k i i g : x = (x ; x ) = (x ; y ) ! x~ = (x ; x~ ) = (±i x ; pk = gkiy ): In another music notation, the flat symbol [ lowers a note by a half step. Similar to this musical notation, the musical isomorphism g[ lowers the indice of the vector space coordinate. The Jacobian matrices of g[ is obtained by [4] µ ¶ µ ¶ ³ ´ M m [ eM @x~ ±j 0 (g¤) = AJ = J s (1. 1) @x y @jgms gmj and the Jacobian matrices of g] is obtained by µ J ¶ µ ¶ ] ¡ ¢ j J @x ±m 0 (g¤) = AM = = js jm : (1. 2) @x~M ps@mg g Let r be an affine connection on M and C r be the complete lift of r. Then C r is an affine connection on TM tangent bundle. We denote by ¡k the components of r affine ¡ ¢ ij connection with respect to the local coordinates xh in M and C ¡k the components of C r ¡ ¢ ij affine connection according to the induced coordinates xh; yh in TM. The g¡lifts are described newly by Cakan and Salimov. The g¡lifts of some tensor fields are obtained by transferring complete lifts of some tensor field from TM tangent bundle to T ¤M cotangent ¤ bundle via the musical isomorphism [8]. In this study we obtained the g¡lift Gr of r affine connection to the T ¤M cotangent bundle via the musical isomorphism. h Let R be curvature tensor of r affine connection with Rkji components on M manifold. ¤ GR the g¡lift of R curvature tensor is obtained to the T ¤M cotangent bundle via the musical isomorphism. 2. THE G-LIFT OF AFFINE CONNECTION Let M be a manifold with r affine connection. There exists a unique C r affine con- nection in TM which satisfies C C C rC Y Z = (rY Z) 1 for any Y; Z 2 =0 (M) [15]. The g-lift of Affine Connection in The Cotangent Bundle 55 ¡ ¢ Let ¡h be components of r according to the local coordinates xh in M. Let C ¡H ji ¡ ¢JI be components of the complete lift C r according to the induced coordinates xh; yh in TM. The components of the C r are given by C ¡h = ¡h ; C ¡h = 0; C ¡h = 0; C ¡h = 0 ji ji ji ji ji (2. 3) C ¡h = ys@ ¡h ; C ¡h = ¡h ; C ¡h = ¡h ; C ¡h = 0 ji s ji ji ji ji ji ji ¡ ¢ ¤ h h C H according to the induced coordinates x ; y in TM [14]. And Let ¡ JI be components ¤ ¡ ¢ C h ¤ of the complete lift r according to the induced coordinates x ; ph in T M. The ¤ components of the C r are given by ¤ ¤ ¤ ¤ C h h C h C h C h ¡ ji = ¡ji; ¡ ji = 0; ¡ ji = 0; ¡ ji = 0 (2. 4) ¤ C h ¡ s s s s t ¢ ¡ ji = ps @h¡ji ¡ @j¡ih ¡ @i¡jh + 2¡ht¡ji ¤ ¤ ¤ C h i C h j C h ¡ ji = ¡¡jh; ¡ ji = ¡¡hi; ¡ ji = 0 ¡ ¢ h ¤ according to the induced coordinates x ; ph in T M [15]. Theorem 2.1. Let M be a n¡dimensional pseudo Riemannian manifold with pseudo Rie- ¤ mannian metric g. Let C r and C r be complete lifts of r affine connection to TM and ¤ T ¤M, respectively. Then the differential of C r by g[, i.e. a g¡lift Gr in the cotangent ¤ bundle T ¤M, coincides with the complete lift C r in the cotangent bundle T ¤M if r is a Riemannian connection which is a metric connection with vanishing torsion. eK @2xK Proof. Using (1:1) and (1:2) from AJI = @xJ @xI equation we obtain the components ek ek ek ek Aji = 0; Aji = 0; Aji = 0; Aji = 0 (2. 5) p @2gks Aek = s ; Aek = @ gki; Aek = @ gkj; Aek = 0: ji @xj@xi ji j ji i ji ¤ Using (1:1) ; (1:2) ; (2:3) and (2:5) the components of Gr are obtained from equation µ ¶ ¤ [ C G H eM eS H C K H eK g¤ ¡ = ¡ JI = AJ AI AK ¡MS + AK AJI where H; I; ::: = 1; :::; 2n: ¤ G¡ h = AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k ji j i k ms j i k ms j i k ms j i k ms +AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k j i k ms j i k ms j i k ms j i k ms +AhAek + AhAek k ji k ji m s h k h = ±j ±i ±k ¡ms = ¡ji 56 Rabia Cakan and Esen Kemer ¤ G¡h = AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k ji j i k ms j i k ms j i k ms j i k ms k +AemAesAh C ¡ +AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k j i k ms j i k ms j i k ms j i k ms +AhAek + AhAek k ji k ji = 0 ¤ G¡h = AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k ji j i k ms j i k ms j i k ms j i k ms h em es h C k em es h C k em es C k em es h C k +A Ai A ¡ + A Ai A ¡ + A Ai A ¡ +A Ai A ¡ j k ms j k ms j k ms j k ms +AhAek + AhAek k ji k ji = 0 G¡h = AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k ji j i k ms j i k ms j i k ms j i k ms +AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k j i k ms j i k ms j i k ms j i k ms +AhAek + AhAek k ji k ji m si k ki = ±j g ghk¡ms + ghk@jg si 1 ki = g 2 (@jgsh + @sghj ¡ @hgjs) ¡ g @jghk 1 si = 2 g (¡@jgsh ¡ @hgjs + @sghj) 1 si = ¡ 2 g (@jgsh + @hgjs ¡ @sghj) i = ¡¡jh ¤ G¡h = AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k ji j i k ms j i k ms j i k ms j i k ms +AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k j i k ms j i k ms j i k ms j i k ms +AhAek + AhAek k ji k ji mj s k kj = g ±i ghk¡ms + ghk@ig mj k kj = g ghk¡mi ¡ g @ighk kj m kj = g ghm¡ki ¡ g @ighk kj 1 kj = g 2 (@kghi + @ighk ¡ @hgki) ¡ g @ighk 1 kj = ¡ 2 g (@hgki + @ighk ¡ @kghi) j = ¡¡hi ¤ G¡h = AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k ji j i k ms j i k ms j i k ms j i k ms +AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k j i k ms j i k ms j i k ms j i k ms +AhAek + AhAek k ji k ji = 0 The g-lift of Affine Connection in The Cotangent Bundle 57 ¤ G¡h = AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k ji j i k ms j i k ms j i k ms j i k ms +AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k j i k ms j i k ms j i k ms j i k ms +AhAek + AhAek k ji k ji = 0 ¤ G¡h = AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k ji j i k ms j i k ms j i k ms j i k ms +AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k + AemAesAh C ¡k j i k ms j i k ms j i k ms j i k ms +AhAek + AhAek k ji k ji m s t k m s t k mt s k = ±j ±i y @kgth¡ms + ±j ±i ghky @t¡ms + pt@jg ±i ghk¡ms m st k 2 tk +±j pt@ig ghk¡ms + ghkpt@jig t k t k mt k st k = y (@kgth)¡ji + ghky @t¡ji + pt@jg ghk¡mi + pt@ig ghk¡js 2 tk +ghkpt@jig t k t k mt k st k = y (@kgth)¡ji + ghky @t¡ji + pt (@j¡g ) g¢hk¡mi + pt (@ig ) ghk¡js t t sk k ts k ts ¡pt (@j¡ih) ¡ ghkpt¡is@jg ¡ ghkpt @j¡is g ¡ ghkpt¡is@jg t t k t k st k = ¡pt (@j¡ih) + y (@kgth¡)¡ji +¢ ghky @t¡ji + pt (@ig ) ghk¡js t sk k ts ¡ghkpt¡is@jg ¡ ghkpt @j¡is g ¡ ¢ t t k t k k ts = ¡pt (@j¡ih) + y (@kgth)¡ji¡+ ghk¢y @t¡ji ¡ ghkpt @j¡is g st k t sk +pt (@ig ) ghk¡js ¡ ghkpt¡is @jg ¡ ¢ t t k ts k ts k = ¡pt (@j¡ih) + y (@kgth)¡ji + ghk g ps@t¡ ji ¡ ptg @j¡is ¢ s mt t sm k t s mk k ms +pt (¡¡img
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