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 obtained 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, tangent 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 G∇ 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 = ∪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 to the basis ∂/∂xi, i.e. y = yi ∂ ∈ T M. The cotangent bundle on M is denoted by x ∂xi x ¡ ¢ ∗ ∗ ∗ 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 ,

53 54 Rabia Cakan and Esen Kemer

i ∗ i.e. px = pidx ∈ Tx M. 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] is 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) where δ 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 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 ). 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˜ δj 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 δm 0 (g∗) = AM = = js jm . (1. 2) ∂x˜M ps∂mg g Let ∇ be an affine connection on M and C ∇ be the complete lift of ∇. Then C ∇ is an affine connection on TM tangent bundle. We denote by Γk the components of ∇ affine ¡ ¢ ij connection with respect to the local coordinates xh in M and C Γk the components of C ∇ ¡ ¢ 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 G∇ of ∇ affine connection to the T ∗M cotangent bundle via the musical isomorphism. h Let R be curvature tensor of ∇ 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. THEG-LIFTOF AFFINE CONNECTION Let M be a manifold with ∇ affine connection. There exists a unique C ∇ affine connection in TM which satisfies

C C C ∇C Y Z = (∇Y Z) 1 for any Y,Z ∈ =0 (M) [15]. The g-lift of Afﬁne Connection in The Cotangent Bundle 55 ¡ ¢ Let Γh be components of ∇ according to the local coordinates xh in M. Let C ΓH ji ¡ ¢JI be components of the complete lift C ∇ according to the induced coordinates xh, yh in TM. The components of the C ∇ are given by

C Γh = Γh , C Γh = 0, C Γh = 0, C Γh = 0 ji ji ji ji ji (2. 3) C Γh = ys∂ Γh , C Γh = Γh , C Γh = Γh , C Γh = 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 Γ JI be components ∗ ¡ ¢ C h ∗ of the complete lift ∇ according to the induced coordinates x , ph in T M. The ∗ components of the C ∇ are given by

∗ ∗ ∗ ∗ C h h C h C h C h Γ ji = Γji, Γ ji = 0, Γ ji = 0, Γ ji = 0 (2. 4) ∗ C h ¡ s s s s t ¢ Γ ji = ps ∂hΓji − ∂jΓih − ∂iΓjh + 2ΓhtΓji ∗ ∗ ∗ C h i C h j C h Γ ji = −Γjh, Γ ji = −Γhi, Γ 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 ∇ and C ∇ be complete lifts of ∇ afﬁne connection to TM and ∗ T ∗M, respectively. Then the differential of C ∇ by g[, i.e. a g−lift G∇ in the cotangent ∗ bundle T ∗M, coincides with the complete lift C ∇ in the cotangent bundle T ∗M if ∇ 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 G∇ are obtained from equation µ ¶ ∗ [ C G H eM eS H C K H eK g∗ Γ = Γ JI = AJ AI AK ΓMS + AK AJI where H, I, ... = 1, ..., 2n:

∗ GΓh = AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk ji j i k ms j i k ms j i k ms j i k ms k +AemAesAh C Γ +AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk j i k ms j i k ms j i k ms j i k ms +AhAek + AhAek k ji k ji = 0

∗ GΓh = AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk 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 Γ + A Ai A Γ + A Ai A Γ +A Ai A Γ j k ms j k ms j k ms j k ms +AhAek + AhAek k ji k ji = 0

GΓh = AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk ji j i k ms j i k ms j i k ms j i k ms +AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk 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 = δj g ghkΓms + 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 = −Γjh

∗ GΓh = AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk ji j i k ms j i k ms j i k ms j i k ms +AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk 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 δi ghkΓms + ghk∂ig mj k kj = g ghkΓmi − g ∂ighk kj m kj = g ghmΓki − g ∂ighk kj 1 kj = g 2 (∂kghi + ∂ighk − ∂hgki) − g ∂ighk 1 kj = − 2 g (∂hgki + ∂ighk − ∂kghi) j = −Γhi

∗ GΓh = AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk ji j i k ms j i k ms j i k ms j i k ms +AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk + AemAesAh C Γk 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 = δj δi y ∂kgthΓms + δj δi ghky ∂tΓms + pt∂jg δi ghkΓms m st k 2 tk +δj pt∂ig ghkΓms + ghkpt∂jig t k t k mt k st k = y (∂kgth)Γji + ghky ∂tΓji + pt∂jg ghkΓmi + pt∂ig ghkΓjs 2 tk +ghkpt∂jig t k t k mt k st k = y (∂kgth)Γji + ghky ∂tΓji + pt (∂j¡g ) g¢hkΓmi + pt (∂ig ) ghkΓjs t t sk k ts k ts −pt (∂jΓih) − ghkptΓis∂jg − ghkpt ∂jΓis g − ghkptΓis∂jg t t k t k st k = −pt (∂jΓih) + y (∂kgth¡)Γji +¢ ghky ∂tΓji + pt (∂ig ) ghkΓjs t sk k ts −ghkptΓis∂jg − ghkpt ∂jΓis g ¡ ¢ t t k t k k ts = −pt (∂jΓih) + y (∂kgth)Γji¡+ ghk¢y ∂tΓji − ghkpt ∂jΓis g st k t sk +pt (∂ig ) ghkΓjs − ghkptΓis ∂jg ¡ ¢ t t k ts k ts k = −pt (∂jΓih) + y (∂kgth)Γji + ghk g ps∂t¡Γji − ptg ∂jΓis ¢ s mt t sm k t s mk k ms +pt (−Γimg − Γimg ) ghkΓjs − ghkptΓ¡is −Γjmg ¢− Γjmg t t k ts k k = −pt (∂¡jΓih) + y (∂kgth)Γji + g¢hkg ps ∂¡tΓji − ∂jΓit ¢ t k ms s k tm s t mk s k mt −ghkps ΓimΓjtg + ΓimΓjtg + ghkps ΓitΓjmg +¡ ΓitΓjmg ¢ t ts m m k ts k k = −pt (∂¡jΓih) + g ps (Γktgmh +¢ Γkhgtm)Γ¡ji + ghkg ps ∂tΓji − ∂jΓ¢it t k ms s k tm s t mk s k mt −ghkps ΓimΓjtg + ΓimΓjtg + ghkps ΓitΓjmg +¡ ΓitΓjmg ¢ t s k ts m k ts k k = −pt (∂¡jΓih) + psΓhkΓji + g gmh¢ psΓktΓji¡ + ghkg ps ∂tΓji − ∂jΓit¢ t k ms s k tm s t mk s k mt −ghkps ΓimΓjtg + ΓimΓjtg + ghkps ΓitΓjmg + ΓitΓjmg t t +ptR − ptR ³hij hij ´ ¡ ¢ s s s s k ts k k = ps ∂hΓij − ∂iΓjh − ∂jΓih + 2ΓhkΓij +ghkg ps ∂tΓji − ∂jΓit t ms t k mt s k −ptRhij − ghkg psΓimΓjt − ghkg psΓimΓjt +g gmtp Γs Γt + g gtsp ΓmΓk hk³ s it jm mh s kt ji ´ ¡ ¢ = p ∂ Γs − ∂ Γs − ∂ Γs + 2Γs Γk − p Rt + g gmsp Γk Γt − Γt Γk s h ij¡ i jh j ih ¢ hk ij t hij hk s mt ji im jt mt s k s k ms k ms k +ghkg ps ΓitΓjm − ΓimΓjt + ghkg ps∂mΓji − ghkg ps∂jΓim 58 Rabia Cakan and Esen Kemer ³ ´ = p ∂ Γs − ∂ Γs − ∂ Γs + 2Γs Γk − p Rt s h ij¡ i jh j ih hk ij t ¢hij ms k k k t t k +ghkg ps ¡∂mΓji − ∂jΓim +¢ ΓmtΓji − ΓimΓjt mt s k s k +ghk³g ps ΓitΓjm − ΓimΓjt ´ = p ∂ Γs − ∂ Γs − ∂ Γs + 2Γs Γk − p Rt + p Rs s ³ h ij i jh j ih hk ij ´ t hij s hji = p ∂ Γs − ∂ Γs − ∂ Γs + 2Γs Γk − ytg Rs + ytg R s s ³ h ij i jh j ih hk ij ´ ts hij ts jih = p ∂ Γs − ∂ Γs − ∂ Γs + 2Γs Γk + yt(−R + R ) s ³ h ij i jh j ih hk ij ´ hijt tjih = p ∂ Γs − ∂ Γs − ∂ Γs + 2Γs Γk + (−R + R ) s ³ h ij i jh j ih hk ij ´ hijt ihtj = p ∂ Γs − ∂ Γs − ∂ Γs + 2Γs Γk + (−R + R ) s ³ h ij i jh j ih hk ij ´ hijt hijt s s s s k = ps ∂hΓij − ∂iΓjh − ∂jΓih + 2ΓhkΓij

∗ We have seen that the components of the complete lift C ∇ to the cotangent bundle T ∗M are given in the form (2.4) according to the induced coordinates in T ∗M. So we obtain

∗ ∗ G∇ = C ∇. ¤

3. THEG-LIFTOF CURVATURE TENSOR Let R be a curvature tensor of ∇ afﬁne connection on M manifold. The components h Rkji of R are given by

h h h h t h t Rkji = ∂kΓji − ∂jΓki + ΓktΓji − ΓjtΓki . There exists a complete lift of curvature tensor C R in TM which satisﬁes

¡ ¢ C R C X, C Y C Z = C (R (X,Y ) Z)

1 for any X,Y,Z ∈ =0 (M). ¡ ¢ C M C h h Let RTSN be components of R according to the induced coordinates x , y in TM. The components of the C R are given by

C m m C m s m Rtsn = Rtsn, Rtsn = y ∂sRtsn, (3. 6) C m m C m m C m m Rtsn = Rtsn, Rtsn = Rtsn, Rtsn = Rtsn ¡ ¢ according to the induced coordinates xh, yh in TM. And the other components are zero. ∗ ∗ ¡ ¢ C M C h Let R TSN be components of R according to the induced coordinates x , ph in ¡ ¢ ∗ ∗ h ∗ C T M. According to the induced coordinates x , ph in T M, the components of the R are given by The g-lift of Afﬁne Connection in The Cotangent Bundle 59

∗ C m m R tsn = Rtsn, ∗ C m a a R tsn = pa(∇mRtsn − ∇nRtsm (3. 7) a b a b a b a b +ΓmbRtsn + ΓtbRnms + ΓsbRmnt + ΓnbRtsm) ∗ ∗ ∗ C m n C m s C m t R tsn = −Rtsm, R tsn = −Rtmn, R tsn = −Rmsn where ∇∂m = ∇m. And the other components are zero [15]. Theorem 3.1. 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 curvature tensor 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 if ∇ is a Riemannian connection which is a metric connection with vanishing torsion. ∗ Proof. Using (1.1) , (1.2) and (3.6) the components of GR are obtained from equation µ ¶ ∗ ³ ´ [C G H eH T S N C M g∗ R = R KJI = AM AK AJ AI R TSN where H, I, ... = 1, ..., 2n:

∗ G h eh t s n C m eh t s n C m eh t s n C m R kji = AmAk Aj Ai Rtsn + AmAk Aj Ai Rtsn + AmAk Aj Ai Rtsn eh t s n C m eh t s n C m eh t s n C m +AmAk Aj Ai Rtsn + AmAk Aj Ai Rtsn + AmAk Aj Ai Rtsn eh t s n C m eh t s n C m eh t s n C m +AmAk Aj Ai Rtsn + AmAk Aj Ai Rtsn + AmAk Aj Ai Rtsn eh t s n C m eh t s n C m eh t s n C m +AmAk Aj Ai Rtsn + AmAk Aj Ai Rtsn + AmAk Aj Ai Rtsn eh t s n C m eh t s n C m eh t s n C m +AmAk Aj Ai Rtsn + AmAk Aj Ai Rtsn + AmAk Aj Ai Rtsn eh t s n C m +AmAk Aj Ai Rtsn h t s n m = δmδkδj δi Rtsn h = Rkji

∗ h ∗ h G G Rkji = 0, Rkji = 0

∗ ∗ G h j G h i R kji = Rkhi, R kji = Rkjh

∗ ∗ ∗ ∗ G h G h G h G h R kji = 0, R kji = 0, R kji = 0, R kji = 0, ∗ ∗ ∗ G h G h G h R kji = 0, R kji = 0, R kji = 0 . ∗ We have seen that the components of the complete lift C R to the cotangent bundle T ∗M is given in the form (3.7) according to the induced coordinates in T ∗M. So we obtain

∗ ∗ GR = C R. ¤

4. CONCLUSION In this paper, we have studied the g-lift of afﬁne connection and curvature tensor in the ∗ ∗ T ∗M cotangent bundle. Obtained the g−lift G∇ and compared with the complete lift C ∇ ∗ in the T ∗M cotangent bundle. Obtained the g−lift GR and compared with the complete ∗ ∗ ∗ lift C R in the T ∗M cotangent bundle. It was seen that the g−lifts G∇ and GR coincide ∗ ∗ with the complete lifts C ∇ and C R, respectively, if ∇ is a Riemannian connection.

5. ACKNOWLEDGMENTS The authors are grateful to the referees and editor for their valuable comments and sug- gestions.

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