Some results on a cross-section in the tensor bundle

Transcription

1 Hacettepe Journa of Matematcs and Statstcs Voume , Some resuts on a cross-secton n te tensor bunde ydın Gezer and Murat tunbas bstract Te present paper s devoted to some resuts concernng wt te compete fts of an amost compex structure and a connecton n a manfod to ts, q-tensor bunde aong te correspondng cross-secton. Keywords: most compex structure, most anaytc tensor, Compete ft, Connecton, Tensor bunde. 2 MS Cassfcaton: Prmary 53C15; Secondary 53B5. 1. Introducton Te beavour of te fts of tensor feds and connectons on a manfod to ts dfferent bundes aong te correspondng cross-sectons are studed by severa autors. For te case tangent and cotangent bundes, see [13, 14, 15] and aso tangent bundes of order 2 and order r, see [3, 11]. In [2], te frst autor and s coaborator studed te compete ft of an amost compex structure n a manfod on te so-caed pure cross-secton of ts p, q-tensor bunde by means of te Tacbana operator for dagona ft to te p, q-tensor bunde see [1] and for te, q-tensor bunde see [5]. Moreover tey proved tat f a manfod admts an amost compex structure, ten so does on te pure cross-secton of ts p, q- tensor bunde provded tat te amost compex structure s ntegrabe. In [6], te autors gve detaed descrpton of geodescs of te p, q- tensor bunde wt respect to te compete ft of an affne connecton. Te purpose of te present paper s two-fod. Frsty, to sow te compete ft of an amost compex structure n a manfod to ts, q-tensor bunde aong te correspondng cross-secton, wen restrcted to te cross-secton determned by an amost anaytc tensor fed, s an amost compex structure. Fnay, to study te beavor of te compete ft of a connecton on te cross-secton of te, q-tensor bunde. taturk Unversty, Facuty of Scence, Department of Matematcs, 2524, Erzurum- TUKEY, Ema: agezer@ataun.edu.tr Correspondng utor. Erzncan Unversty, Facuty of Scence and rt, Department of Matematcs, 243, Erzncan-TUKEY, Ema: matunbas@erzncan.edu.tr

2 Trougout ts paper, a manfods, tensor feds and connectons are aways assumed to be dfferentabe of cass C. so, we denote by I p qm te set of a tensor feds of type p, q on M, and by I p qt q M te correspondng set on te, q -tensor bunde T q M. Te Ensten summaton conventon s used, te range of te ndces, j, s beng aways {1, 2,..., n}. 2. Premnares Let M be a dfferentabe manfod of cass C and fnte dmenson n. Ten te set Tq M = P M Tq P, q >, s te tensor bunde of type, q over M, were denotes te dsjont unon of te tensor spaces Tq P for a P M. For any pont P of Tq M suc tat P Tq M, te surjectve correspondence P P determnes te natura projecton π : Tq M M. Te projecton π defnes te natura dfferentabe manfod structure of Tq M, tat s, Tq M s a C -manfod of dmenson n n q. If x j are oca coordnates n a negborood U of P M, ten a tensor t at P wc s an eement of Tq M s expressbe n te form x j, t j1...j q, were t j1...j q are components of t wt respect to natura base. We may consder x j, t j1...j q = x j, x j = x J, j = 1,..., n, j = n 1,..., n n q, J = 1,..., n n pq as oca coordnates n a negborood π 1 U. Let V = V x and = j1...j q dx j1 dx jq be te oca expressons n U of a vector fed V and a, qtensor fed on M, respectvey. Ten te vertca ft V of and te compete ft C V of V are gven, wt respect to te nduced coordnates, by 2.1 V = and 2.2 j1...j q V j C V = q t j1...m...j q jλ V m. Suppose tat tere s gven a tensor fed ξ I qm. Ten te correspondence x ξ x, ξ x beng te vaue of ξ at x M, determnes a mappng σ ξ : M Tq M, suc tat π σ ξ = d M, and te n dmensona submanfod σ ξ M of Tq M s caed te cross-secton determned by ξ. If te tensor fed ξ as te oca components ξ k1 k q x k, te cross-secton σ ξ M s ocay expressed by 2.3 { x k = x k, x k = ξ k1 k q x k wt respect to te coordnates x k, x k n Tq M. Dfferentatng 2.3 by x j, we see tat n tangent vector feds B j to σ ξ M ave components 2.4 Bj K = xk x j = δj k j ξ k1 k q wt respect to te natura frame { k, k } n T q M.

3 On te oter and, te fbre s ocay expressed by { x k = const., t k1 k q = t k1 k q, t k1 k q beng consdered as parameters. Tus, on dfferentatng wt respect to x j = t j1 j q, we see tat n q tangent vector feds C j to te fbre ave components 2.5 C K j = xk x = j δ j1 k 1 δ jq k q wt respect to te natura frame { } k, k n T q M. We consder n π 1 U Tq M, n n q oca vector feds B j and C j aong ] σ ξ M. Tey form a oca famy of frames [B j, C j aong σ ξ M, wc s caed te adapted B, Cframe of σ ξ M n π 1 U. Takng account of 2.2 on te cross-secton σ ξ M, and aso 2.4 and 2.5, we can easy prove tat, te compete ft C V as aong σ ξ M components of te form 2.6 C V V = j L V ξ j1 j q wt respect to te adapted B, C-frame. From 2.1, 2.4 and 2.5, te vertca ft V aso as components of te form 2.7 V = j1...j q wt respect to te adapted B, C- frame. 3. most compex structures on a pure cross-secton n te, q- tensor bunde tensor fed ξ I qm s caed pure wt respect to ϕ I 1 1M, f [2, 4, 5, 7, 8, 9, 1, 12]: 3.1 ϕ r j 1 ξ r jq = = ϕ r j q ξ j1 r = ξ j1 j q. In partcuar, vector and covector feds w be consdered to be pure. Let I qm denotes a modue of a te tensor feds ξ I qm wc are pure wt respect to ϕ. Now, we consder a pure cross-secton σ ϕ ξ M determned by ξ I qm. Te compete ft C ϕ of ϕ aong te pure cross-secton σ ϕ ξ M to Tq M as oca components of te form C ϕ = ϕ k Φ ϕ ξ k1...k q ϕ r1 k 1 δ r2 k 2...δ rq k q wt respect to te adapted B, Cframe of σ ϕ ξ M, were Φ ϕξ k1 k q = ϕ m m ξ k1 k q ξk1 k q q ka ϕ m ξ k1 m k q s te Tacbana operator. a=1

4 and We consder tat te oca vector feds C X = C x =C δ x = δ V X = V dx 1 dx q = V δ 1 1 δ q q dx 1 dx q = δ 1 1 δ q q = 1,..., n, = n 1,..., n n q span te modue of vector feds n π 1 U. Hence, any tensor feds s determned n π 1 U by ter actons on C V and V for any V I 1 M and I qm. Te compete ft C ϕ aong te pure cross-secton M as te propertes σ ϕ ξ 3.2 { C ϕ C V = C ϕv V L V ϕ ξ, V I 1 M, C ϕ V = V ϕ, I qm, wc caracterze C ϕ, were ϕ I qm. vector fed on Tq M and ocay expressed by V L V ϕ ξ = L V ϕ j 1 ξ j2 q emark tat V L V ϕ ξ s a wt respect to te adapted B, C-frame, were ξ 1 q are oca components of ξ n M [5] Teorem. Let M be an amost compex manfod wt an amost compex structure ϕ. Ten, te compete ft C ϕ I 1 1Tq M, wen restrcted to te pure cross-secton determned by an amost anaytc tensor ξ on M, s an amost compex structure. Proof. If V I 1 M and I qm, n vew of te equatons and of 3.2, we ave 3.3 C ϕ 2 C V = C ϕ 2 C V V N ϕ ξ C V and 3.4 C ϕ 2 V = C ϕ 2 V, were N ϕ,x Y = L ϕx ϕ ϕl X ϕy = [ϕx, ϕy ] ϕ [X, ϕy ] ϕ [ϕx, Y ] ϕ 2 [X, Y ] = N ϕ X, Y s notng but te Njenus tensor constructed by ϕ. Let ϕ I 1 1M be an amost compex structure and ξ I qm be a pure tensor wt respect to ϕ. If Φ ϕ ξ =, te pure tensor ξ s caed an amost anaytc, qtensor. In [4, 7, 9], t s proved tat ξ ϕ I qm s an amost anaytc tensor f and ony f ξ I qm s an amost anaytc tensor. Moreover f ξ I qm s an amost anaytc tensor, ten N ϕ ξ =. Wen restrcted to te pure cross-secton determned by an amost anaytc tensor ξ on M, from 3.3, 3.4 and nearty of te compete ft, we ave C ϕ 2 = C ϕ 2 = C I M = I T q M. Ts competes te proof.

5 4. Compete ft of a symmetrc affne connecton on a crosssecton n te, q-tensor bunde We now assume tat s an affne connecton wt zero torson on M. Let Γ j be components of. Te compete ft C of to Tq M as components C Γ I MS suc tat 4.1 C Γ ms = Γ ms, C Γms = C Γ ms = C Γ ms = C Γ ms =, C Γ ms = Γ sc m c δ s1 1...δ sc1 c1 δ sc1 c1...δ sq q, C Γ ms = C Γ ms = c=1 Γ mc s c δ m1 1 c=1...δ mc1 c1 δ mc1 c1...δ mq q, m Γ a s c Γ r m c Γ a sr Γ r msγ a r c t 1... c1a c1... q c=1 1 Γ m 2 c Γ r s b Γ m b Γ r s c t 1... b1 r b1... c1 c1... q b=1 c=1 t 1... q d=1 d km wt respect to te natura frame n Tq M, were δj Kronecker deta and km s components of te curvature tensor of [6]. We now study te affne connecton nduced from C on te cross-secton σ ξ M determned by te, qtensor fed ξ n M wt respect to te adapted B, C- frame of σ ξ M. Te vector feds C j gven by 2.5 are neary ndependent and not tangent to σ ξ M. We take te vector feds C j as normas to te cross-secton σ ξ M and defne an affne connecton nduced on te cross-secton. Te affne connecton nduced σ ξ M from te compete ft C of a symmetrc affne connecton n M as components of te form 4.2 Γ j = j B were B are defned by and tus C Γ CBB C j B B B, B, C = B, C B = δ,, C = j ξ k1...k q, δ j1 k 1...δ jq k q. Substtutng 4.1, 2.4, 2.5 and 4.3 n 4.2, we get Γ j = Γ j, were Γ j are components of n M. From 4.2, we see tat te quantty 4.4 j B C Γ CBBj C B B Γ jb

6 s a near combnaton of te vectors C n 4.4 and fnd j ξ 1... q ξ 1... q Hence, representng 4.4 by j B 4.5 j B = j ξ 1... q λ j.. To fnd te coeffcents, we put =, we obtan ξ 1... q λ j C. Te ast equaton s notng but te equaton of Gauss for te cross-secton σ ξ M determned by ξ 1... q. Hence, we ave te foowng proposton Proposton. Te cross-secton σ ξ M n Tq M determned by a, q tensor ξ n M wt symmetrc affne connecton s totay geodesc f and ony f ξ satsfes j ξ 1... q ξ 1... q λ j =. Now, et us appy te operator k to 4.5, we ave 4.6 k j B = k j ξ 1... q ξ 1... q λ j C. ecang tat k j B j k B = DCB Bk D Bj C B B kj B, and usng te cc dentty for a tensor fed of type, q, from 4.6 we get = [ k DCB Bk D Bj C B B λ j kj B j λ k ξ 1... q kj λ ξ 1... q Tus we ave te resut beow. kj ξ 1... q λ j k ξ 1... q 4.2. Proposton. DCB B D f and ony f k j λ j = kj ξ 1... q k Bj C B B λ k ξ 1... q λ k j ξ 1... q. kj λ λ k j ξ 1... q ]C s tangent to te cross-secton σ ξ M ξ 1... q λ j k ξ 1... q.

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