Bulletn of the Translvana Unversty of Braşov Vol 554 No. 2-202 Seres III: Mathematcs Informatcs Physcs 75-88 ABOUT THE GROUP OF TRANSFORMATIONS OF METRICAL SEMISYMMETRIC N LINEAR CONNECTIONS ON A GENERALIZED HAMILTON SPACE OF ORDER TWO Monca PURCARU and Mrela TÂRNOVEANU 2 Abstract In the present paper we obtan n a generalzed Hamlton space of order two the transformaton laws of the torson and curvature tensor felds wth respect to the transformatons of the group T N of the transformatons of N lnear connectons havng the same nonlnear connecton N. We also determne n a generalzed Hamlton space of order two the set of all metrcal semsymmetrc N lnear connectons n the case when the nonlnear connecton s fxed and prove that ths set ms T N of the transformatons of metrcal semsymmetrc N lnear connectons havng the same nonlnear connecton N together wth the composton of mappngs s a group. We obtan some mportant nvarants of the group ms T N and gve ther propertes. We also study the transformatons laws of the torson d tensor felds wth respect to the transformaton of the group ms T N. 2000 Mathematcs Subject Classfcaton: 53B05 Key words: second order cotangent bundle generalzed Hamlton space of order two nonlnear connecton N lnear connecton metrcal N lnear connecton metrcal semsymmetrc N lnear connecton transformatons group subgroup torson curvature nvarants. Introducton Dfferental geometry of the second order cotangent bundle T 2 M π 2 M was ntroduced and studed by R. Mron [6] R. Mron H. Shmada D. Hrmuc V.S. Sabău [8] and Gh. Atanasu and M. Târnoveanu []. Ths geometry s based on the dfferental geometry of the cotangent bundle see also: Gh. Atanasu [2] S. Ianuş [3] R. Mron [5] C. Udrşte [0]. In the present secton we keep the general settng from R. Mron H. Shmada D. Hrmuc V.S. Sabău [8] and subsequently we recall only some needed notons. For more detals see [8]. Translvana Unversty of Braşov Faculty of Mathematcs and Informatcs Iulu Manu 50 Braşov 50009 Romana e-mal: mpurcaru@untbv.ro 2 Translvana Unversty of Braşov Faculty of Mathematcs and Informatcs Iulu Manu 50 Braşov 50009 Romana e-mal: m tarnoveanu@yahoo.com
76 Monca Purcaru and Mrela Târnoveanu Let M be a real n dmensonal manfold and let T 2 M π 2 M be the dual of the 2 tangent bundle or 2 cotangent bundle. A pont u T 2 M can be wrtten n the form u = x y p havng the local coordnates x y p = 2... n. A change of local coordnates on the 3n dmensonal manfold T 2 M s x = x x... x n det x x j 0 ȳ = x x j y j p = xj x p j j = 2... n. We denote by T 2 M = T 2 M \ 0 where 0 : M T 2 M s the null secton of the projecton π 2. Let us consder the tangent bundle of the dfferentable manfold T 2 M T T 2 M τ 2 T 2 M where τ 2 s the canoncal projecton and the vertcal dstrbuton V : u T 2 M V u T u T 2 M locally generated by the vector felds u u y p u T 2 M. The followng F T 2 M lnear mappng J : χ T 2 M χ T 2 M defned by: J. x = y J y = 0 J = 0 u p T 2 M.2 s a tangent structure on T 2 M. We denote wth N a nonlnear connecton on the manfold T 2 M wth the local coeffcents N j x y p N j x y p j = 2... n. Hence the tangent space of T 2 M n the pont u T 2 M s gven by the drect sum of vector spaces: T u T 2 M = N u W u W 2 u u T 2 M..3 A local adapted bass to the drect decomposton.3 s gven by: δ δx y = 2... n.4 p where: δ δx = x N j y j + N j..5 p j Wth respect to the coordnates transformatons. we have the rules: δ δx = xj δ x δ x j ; y = xj x ȳ j ; = x p x j..5 p j The dual bass of the adapted bass.4 s gven by: δx δy δp.6 where: δx = dx δy = dy + N jdx j δp = dp N j dx j..6
About the group of transformatons of metrcal semsymmetrc N lnear connectons 77 Wth respect to. the covector felds.6 are transformed by the rules: δ x = x x j δxj δȳ = x x j δyj δ p = xj x δp j..6 Let D be an N lnear connecton on T 2 M wth the local coeffcents n the adapted bass: DΓ N = H jk C jk C jk. An N lnear connecton D s unquely represented n the adapted bass.4 n the followng form: δ D δ = H k δx j δx j δ D δx k δ = H k δx j y j D y k δ δx j p = H kj δ D = C k y j δx j δ D δx k = C k y j y j D y k y j p = C kj.7 D p j δ = C kj δ δx D δx k p j = C kj y D y k p j p = C j k. 2 The transformatons of the d tensors of torson and curvature In the followng we shall study the Abelan group T N. Its elements are the transformatons t : DΓ N D Γ N gven by see[9]: N j = N j N j = N j H k j = H k j B k j C k j = C k j D k j C kj = C kj D kj j k = 2... n. 2. Frstly we shall study the transformatons of the d tensors of torson of DΓ N. Proposton 2.. The transformatons of the Abelan group T N gven by2. lead to the transformatons of the d tensors of torson n the followng way: R jk = R jk jk = R jk R2 B j k = B j k jk = B jk 2.2 2 B2 2 T jk = T jk + B kj B jk 2.3 S jk = S jk + D kj D jk S jk = S jk + D kj jk D 2.4 P jk = P jk + B kj P jk = P jk B jk. 2.5 2 2 Proof. Usng 7.2 p.256 [8] 6.3 6.3 6.3 p.273 [8] and 2. we have the results.
78 Monca Purcaru and Mrela Târnoveanu Now we shall study the transformatons of the d tensors of curvature of DΓ N see 6.4 -p.274 [8] and 5.2 -p.270 [8] by a transformaton 2.. We get: Proposton 2.2. The transformatons of the Abelan group T N gven by 2. lead to the transformatons of the d tensors of curvature n the followng way: R h jk = R h jk D hm R m jk D m h R mjk B hmt m jk+ 2 +A jk B m hj B mk B 2.6 hj k P h jk = P h jk D hm P m jk D m h B mjk B hmc m jk+ 2 +B m hjd mk D m hkb mj B hj k +D hk j P k h j = P k h j D m k h P mj D hm B j mk B hmc mk j + 2 +B m hjd k m D mk h B mj B hj k +D k h j 2.7 2.8 S h jk = S h jk D hms m jk + A jk D hj k +D m hjd mk 2.9 S h j k = S h j k D hj k +D h k j C j mk D hm C k mjd h m + D m hjd m k D h mk D mj 2.0 S h jk = S h jk + D h m S m jk + A jk D h j k +D h mj D m k 2. where A j denotes the alternate summaton and m m and m denote the h-covarant dervatve the w -covarant dervatve and the w 2 -covarant dervatve wth respect to DΓN respectvely. We shall consder the tensor felds: K h jk = R h jk C hm R m jk C h m R 2 mjk 2.2 P h jk = A jk P h jk C N m j hm y k P k h j = A jk P k h j C N m j hm m + C N jm h + C h m N jm 2.3. 2.4 Proposton 2.3. By a transformaton of the Abelan group T N gven by 2. the tensor felds K h jk P h jk P h j k are transformed accordng to the followng laws: K h jk = K h jk B hmt m jk + A jk B m hjb mk B hj k 2.5
About the group of transformatons of metrcal semsymmetrc N lnear connectons 79 P h jk = P h jk D hmt m jk B hms m jk+ +A jk B hj k +D hk j + B m hjd mk D m hkb mj P k h j = P k h j + A jk B hj k +D k h j D m h H k mj B hmc mk j + B m hjd k m D mk h B mj. 2.6 2.7 Proof. From 2.7 we get: A jk P h jk = A jk P h jk + A jk D m hm P jk D m h B mjk 2 A jk B hmc m jk + A jk B m hjd mk D m hkb mj B hj k +D hk j. Usng 7.2 p.256 [8] 6.3 6.3 p.273 [8] and 2.8 we have: A jk P h jk = A jk P h jk + A jk C hm C hm + Ch m C h m N jm B hms m jk+ +A jk B hj k +D hk j + B m hjd mk D m hkb mj N m j. y k H m kj + If we separate the terms we get: A jk P h jk C N m j hm y k = A jk P h jk C N m j hm y k + C h m N jm + C m N jm h = D hmt m jk B hms m jk+ +A jk B hj k +D hk j +B m hjd mk D m hkb mj Usng 2.3 we obtan: 2.6. Analogous we obtan the other formulas. 3 Metrcal semsymmetrc N lnear connectons n GH 2n spaces Defnton 3.. [8] A generalzed Hamlton space of order two s a par GH 2n = M g j x y p where: g j s a d tensor feld of type 2 0 symmetrc and nondegenerate on the manfold T 2 M. 2 The quadratc form g j X X j has a constant sgnature on T 2 M. g j s called the fundamental tensor or metrc tensor of space GH 2n..
80 Monca Purcaru and Mrela Târnoveanu In the case when T 2 M s a paracompact manfold then on T 2 M the metrc tensors g j x y p exst postvely defned such that M g j s a generalzed Hamlton space. Defnton 3.2. [8] A generalzed Hamlton metrc g j x y p of order two on short GH metrc s called reductble to an Hamlton metrc H metrc of order two f there exsts a functon H x y p on T 2 M such that: g j = 2 H. 3. 2 p p j The covarant tensor feld g j s obtaned from the equatons g j g jk = δ k. 3.2 g j s a symmetrc nondegenerate and covarant of order two d tensor feld. If a nonlnear connecton N wth the coeffcents N j x y p N j x y p s a pror gven let us consder the drect decomposton.3 and the adapted bass to t.4 where.5 hold. The dual adapted bass s.6 where.6 hold. An N lnear connecton: DΓ N = H jk C jk C jk determnes the h w w 2 covarant dervatves n the tensor algebra of d tensor felds. Defnton 3.3. [8] An N lnear connecton DΓ N s called metrcal wth respect to GH metrc g j f g j s covarant constant or absolute parallel wth respect to DΓ N.e. The tensoral equatons 3.3 mply: g j k = 0 g j k = 0 g j k = 0. 3.3 g j k = 0 g j k = 0 g j k = 0. 3.4 Theorem 3.. [8]. There s a unque N lnear connecton D Γ N = = H jk C jk C jk havng the propertes:. The nonlnear connecton s a pror gven. 2. D Γ N s metrcal wth respect to GH metrc g j.e.3.3 are verfed. 3. The torson tensors T jk S jk and S jk vansh. 2. The prevous connecton has the coeffcents C jk and C jk gven by C jk = 2 gm gmk C jk = 2 g m y j g mk p j + g jm y k + gjm g jk y m gjk p m 3.5 and H jk are generalzed Chrstoffel symbols: H jk = 2 gm δgmk δx j + δg jm δx k δg jk δx m. 3.6
About the group of transformatons of metrcal semsymmetrc N lnear connectons 8 The Obata s operators are gven by: Ω j hk = δh 2 δj k g hkg j Ω j hk = δh 2 δj k + g hkg j. 3.7 There s nferred: Proposton 3.. The Obata s operators have the followng propertes: Ω r sj + Ω r sj = δ sδ r j 3.8 Ω r sjω sn mr = Ω n mj Ω r sjω sn mr = Ω n mj Ω r sjω sn mr = Ω r sjω sn mr = 0 3.9 Ω r rj = Ω r s = 0 Ω r j = 2 n δr j Ω r j = 2 n + δr j. 3.0 Theorem 3.2. [8] There s a unque metrcal connecton DΓ N = H jk C jk C jk wth respect to GH metrc g j havng the torson d tensor felds T jk S jk S jk a pror gven. The coeffcents of DΓ N are gven by the followng formulas: H jk = 2 gm δgmk + δg jm δg jk δx j δx k δx + m + 2 gm g mh T h jk g jh T h mk + g kh T h jm C jk = 2 gm gmk + g jm g jk y j y k y + m + 2 gm g mh S h jk g jh S h mk + g kh S h jm C jk = 2 g m 2 g m g mk p j + gjm gjk p m g mh S h jk g jh S h mk + g kh S h jm. 3. Defnton 3.4. [] An N lnear connecton on T 2 M s called semsymmetrc f: T jk = 2 δ j σ k + δk σ j S jk = δ 2 j τ k + δk τ j jk S = δ j 2 vk + δ k v j 3.2 where σ τ χ T 2 M and v χ T 2 M. Theorem 3.3. The set of all metrcal semsymmetrc N lnear connectons wth local coeffcents DΓ N = H jk C jk C jk s gven by: H jk = H jk + 2 gjk g m σ m + σ j δ k C jk = C jk + 2 gjk g m τ m + τ j δ k 3.3 C jk = C jk + 2 g jk g m v m + v j δ k where D Γ N = H jk C jk C jk are the local coeffcents 3.6 and 3.5 of the metrcal N lnear connecton gven n Theorem 3. and σ τ χ T 2 M and v χ T 2 M. Proof. Usng Theorem 3.2 and Defnton 3.4 we obtan the results by drect calculaton.
82 Monca Purcaru and Mrela Târnoveanu 4 The group of transformatons of metrcal semsymmetrc N lnear connectons Let N be a gven nonlnear connecton on T 2 M. Then any metrcal semsymmetrc N lnear connecton wth local coeffcents DΓ N = H jk C jk C jk s gven by 3. wth 3.2. From Theorem 3.3 we have: Theorem 4.. Two metrcal semsymmetrc N lnear connectons:d and D wth local coeffcents: DΓ N = H jk C jk C jk and DΓ N = = H jk C jk C jk are related as follows: H jk = H jk + 2 gjk g m σ m + σ j δk C jk = C jk + 2 gjk g m τ m + τ j δk 4. C jk = C jk + 2 g jk g m v m + v j δ k where σ τ χ T 2 M and v χ T 2 M. Conversely gven σ τ χ T 2 M and v χ T 2 M the above 4. s thought to be a transformaton of a metrcal semsymmetrc N lnear connecton D wth local coeffcents DΓ N = H jk C jk C jk local coeffcents DΓ N = H jk C jk C jk to a metrcal semsymmetrc N lnear connecton D wth. We shall denote ths transformaton by: t σ τ v. Thus we have: Theorem 4.2. The set ms T N of all transformatons t σ τ v : DΓ N DΓ N of the metrcal semsymmetrc N lnear connectons gven by 4. s an Abelan group together wth the mappng product. Ths group acts on the set of all metrcal semsymmetrc N lnear connectons correspondng to the same nonlnear connecton N transtvely. Theorem 4.3. By means of transformaton 4. the tensor felds: K h jk P h jk P h j k S h jk and S h jk are changed by the laws: K h jk = K h jk + A jk Ω r jh σ rk 4.2 P h jk = P h jk + A jk Ω r jh γ rk 4.3 P k h j = P h jk + A jk Ω r jh σ r k +Ω j rh vr k + Ω m rh H k mjv r + C rk j σ m + 4.4 + 2 Ωk hj σ rv r + 4 δk h g jrg s σ s v r + 4 δ jg rh g ks σ s v r 4 g jsg k σ h v s 4 g jhg rk σ r v
About the group of transformatons of metrcal semsymmetrc N lnear connectons 83 S h jk = S h jk + A jk Ω r jh τ rk 4.5 where: S h jk = S h jk + A jk Ω j rh vrk 4.6 σ rk = σ r σ k + σ r k + 4 g rk σ σ = g rm σ r σ m 4.7 γ rk = σ k τ r + σ r τ k + σ r k +τ r k + 4 g rkγ γ = g rm σ r τ m + σ m τ r 4.8 Proof. Usng 2. and 4. we get: τ rk = τ r τ k + τ r k + 4 g rkτ τ = g rs τ r τ s 4.9 v rk = v r v k + v r k 4 grk v v = g rs v r v s. 4.0 B jk = σj δk 2 + g jkg m σ m = Ω m kj σ m D jk = τj δk 2 + g jkg m τ m = 4. Ω m kj τ m D jk = v j δ k + g jk g m v m = Ω jk m 2 vm. By applyng Proposton 2.3 relatons 3.2 and 4. we obtan the results. Usng these results we can determne some nvarants of the group ms T N. To ths am we elmnate σ j γ j τ j and v j from 4.2 4.3 4.4 and 4.5 and we obtan: Theorem 4.4. For n > 2 the followng tensor felds: H h jk N h jk M h jk M h jk of metrcal semsymmetrc N lnear connectons on T 2 M are nvarants of the group ms T N : H h jk = K h jk + 2 n 2 A jk Ω r jh K rk g rkk 4.2 2 n where: N h jk = P h jk + 2 n 2 A jk M h jk = S h jk + 2 n 2 A jk Ω r jh P rk Ω r jh S rk g rkp 2 n g rks 2 n 4.3 4.4 M jk h = S jk h + 2 n 2 A jk Ω j rh S rk grk S 4.5 2 n K hj = K h j K = g hj K hj P hj = P h j P = g hj P hj S hj = S h j S = g hj S hj S j = S h jh S = g j S j.
84 Monca Purcaru and Mrela Târnoveanu In order to fnd other nvarants of the group ms T N let us consder the transformaton formulas of the torson d tensor felds by a transformaton t σ τ v : DΓ N DΓ N of metrcal semsymmetrc N lnear connectons on T 2 M correspondng to the same nonlnear connecton N gven by 4.. Usng Proposton 2. and transformaton 4. by drect calculaton we obtan: Proposton 4.. By a transformaton 4. of metrcal semsymmetrc N lnear connectons correspondng to the same nonlnear connecton N : t σ τ v : DΓ N DΓ N the torson tensor felds: R jk R 2 jk B j k B jk B 2 jk T jk S jk S jk P jk P 2 jk are transformed as follows: R jk = R jk R jk = R jk 2 2 B jk = B jk B k j = B k j 2 2 B k j = B k j B jk = B jk 2 2 T jk = T jk + 2 A jk σj δk S jk = S jk + 2 A jk τj δk S jk = S jk + 2 A jk v j δ k P jk = P jk + 2 σ k δj + g jkg m σ m jk = P jk 2 σj δk + g jkg m σ m. 2 P 2 4.6 We denote wth: t N j jk = A jk y k t 2 j k = A jk N j t 3 jk = A jk Njk p 4.7 and wth: t jk = Σ jk g m t m jk t k j = Σ jk g m t j mk 2 2 t jk = Σ jk g m t m jk 3 3 T jk = Σ jk g m T m jk R jk = Σ jk g m R m jk C jk = Σ jk g m C m jk 4.8
About the group of transformatons of metrcal semsymmetrc N lnear connectons 85 P jk = Σ jk g m P m jk jk = Σ jk g m P m jk 2 P 2 S jk = Σ jk g m S m jk B jk = Σ jk g m B m jk B jk = Σ jk g m A jk B 2 B jk = Σ jk g m A jk B where Σ jk... denotes the cyclc summaton and wth: K 2 m jk m jk g m P mjk jk = g km T m j + A j K 2 jk = g m T m jk A jk g km H m j 2 K jk = g m S m jk A jk g km C m j 3 K jk = A jk g km 2 P m j + P m j 2 4 K jk = g jm C m k + g m C m jk ϕ jk = A j g m B m jk 2 ϕ jk = A jk g jm A k B m k 3 ϕ jk = A k g jm P m k g km P m j 2. 4.9 4.20 Remark 4.. It s noted that t jk t k j t jk T jk R jk S jk 2 3 B jk 2 B jk alternate K jk ϕ jk are alternate wth respect to j K jk 2 2 K jk K 3 jk ϕ 2 jk are alternate wth respect to j k and ϕ 3 jk s alternate wth respect to k. Theorem 4.5. The tensor felds: R jk R 2 jk B j k B 2 jk t jk t 2 j k t 3 jk t jk t k j 2 t jk T jk R jk C jk P jk P jk S jk B jk B jk 3 2 2 K jk K 3 4 jk K jk ϕ jk ϕ 2 jk ϕ 3 jk are nvarants of the group ms T N. 2 B jk K jk are K 2 jk Proof. By means of transformatons of the torson gven n 4.6 and usng the notatons: 4.7 4.8 4.9 4.20 by drect calculaton from 4. we obtan the results.
86 Monca Purcaru and Mrela Târnoveanu Theorem 4.6. Between the nvarants n Theorem 4.5 the followng relatons exst: Σ jk jk = T K jk + A j P jk = t jk 4.2 Σ jk jk = 0 4.22 K2 Σ jk 2Kjk = 0 4.23 3Kjk Σ jk = 3 2 T jk + 2 t jk + t jk 4.24 3 4Kjk 4Kjk Σ jk = C jk + C kj A j = 0 4.25 Σ jk ϕjk 2ϕjk Σ jk = 2 = B jk B kj = A jk B jk 4.26 B jk B kj = 2A jk B jk 4.27 Σ jk 3ϕjk = P 2 jk P jk + P kj P 2 kj = B jk + 2 B jk. 4.28 Proof. Usng notatons 4.7 4.8 4.9 4.20 Remark 4. and the defntons of the torson d tensor felds gven n [8] - p.256 and - 273 by drect calculatons we obtan the results. References [] Atanasu Gh. and Târnoveanu M. New aspects n the dfferental geometry of the second order cotangent bundle Unv. de Vest dn Tmşoara 90 2005-64. [2] Atanasu Gh. The nvarant expresson of Hamlton geometry Tensor N.S. Japona 47 988 23-32. [3] Ianuş S. On dfferental geometry of the dual of a vector bundle The Proc. of the Ffth Natonal Sem. of Fnsler and Lagrange Spaces Unv. dn Braşov 988 73-80. [4] Matsumoto M. The theory of Fnsler connectons Publ. of the Study Group of Geometry 5 Depart. Math. Okayama Unv. 970 XV + 220pp. [5] Mron R. Hamlton geometry Semnarul de Mecancă Unv. Tmşoara 3 987-54. [6] Mron R. Hamlton spaces of order k grater than or equal to Int. Journal of Theoretcal Phys. 39 2000 no. 9 2327-2336.
About the group of transformatons of metrcal semsymmetrc N lnear connectons 87 [7] Mron R. Ianuş S. and Anastase M. The geometry of the dual of a vector bundle Publ. de l Inst. Math. 4660 989 45-62. [8] Mron R. Hrmuc D. Shmada H. and Sabău V.S. The geometry of Hamlton and Lagrange spaces Kluwer Academc Publsher FTPH 8 200. [9] Purcaru M.A.P. and Târnoveanu M. On transformaton group of N-lnear connectons one second order cotangent bundle to appear. [0] Saunders D.J. The geometry of jet bundles Cambrdge Unv. Press 989. [] Udrşte C. Şandru O. Dual nonlnear connectons Proc. of 22 nd Conference Dfferental Geometry and Topology Polytechnc Insttute of Bucharest Romana sept. 99. [2] Yano K. Ishhara S. Tangent and cotangent bundles. Dfferental Geometry. M. Dekker Inc. New-York 973.
88 Monca Purcaru and Mrela Târnoveanu