About Three Important Transformations Groups

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1 About Tree Important Transformatons Groups MONICA A.P. PURCARU Translvana Unversty of Braşov Department of Matematcs Iulu Manu Street Braşov ROMANIA m.purcaru@yaoo.com MIRELA TÂRNOVEANU Translvana Unversty of Braşov Department of Matematcs Iulu Manu Street Braşov ROMANIA m tarnoveanu@yaoo.com LAURA CIUPALĂ Translvana Unversty of Braşov Department of Computer Scence Iulu Manu Street Braşov ROMANIA laura cupala@yaoo.com Abstract: In te present paper e ntroduce te concepts of conformal metrcal d-structure and of conformal metrcal N-lnear connecton t respect to te conformal metrcal d-structure correspondng to an 1-form on a generalzed Hamlton space. We determne te set of all conformal metrcal N-lnear connectons n te case en te nonlnear connecton s arbtrary and e fnd mportant examples and partcular cases. We fnd te transformatons group of tese connectons. We study te role of te torson d-tensor felds T S and S n ts teory especally n te determnaton of te set of all semsymmetrc conformal metrcal N-lnear connectons t respect to te conformal metrcal d-structure correspondng to te same nonlnear connecton N. We gve te transformatons group of tese connectons and oter to mportant groups and e fnd ter remarable nvarants. Fnally e determne te set of all metrcal N-lnear connectons n te case en te nonlnear connecton s arbtrary e gve mportant examples and partcular cases and for te case en te nonlnear connecton s fxed e fnd te transformatons group of tese connectons. Key Words: second order cotangent bundle nonlnear connecton N-lnear connecton metrcal d-structure conformal metrcal d-structure conformal metrcal N-lnear connecton semsymmetrc conformal metrcal N-lnear connecton metrcal N-lnear connecton transformatons group subgroup nvarants. 1 Introducton Te geometry of te cotangent bundle T M π M as been studed by R.Mron S.Watanabe and S.Yeda n [5] by K.Yano and S.Isara n [14] by R.Mron D.Hrmuc H.Smada and S.Sabău n [4] C. Udrşte and O. Şandru n [12] etc. Contrbutons n development of ts teory ad also: Te dfferental geometry of te second order cotangent bundle T 2 M π 2 M as ntroduced and studed by R. Mron n [2] R. Mron D. Hrmuc H. Smada V.S. Sabău n [4] G. Atanasu and M. Târnoveanu n [1] C. Udrşte M.Popescu and P.Popescu n [11] C. Udrşte n [8] [9] C. Udrşte D. Oprş n [1] C. Udrşte I. Tevy n [13] etc. In te present secton e eep te general settng from R. Mron D. Hrmuc H. Smada V.S. Sabău [4] and subsequently e recall only some needed notons. For more detals see [4]. Let M be a real n dmensonal manfold and let T 2 M π 2 M be te dual of te 2 tangent bundle or 2 cotangent bundle. A pont u T 2 M can be rtten n te form u = x y p avng te local coordnates x y p = n. A cange of local coordnates on te 3n dmensonal manfold T 2 M s x = x x 1... x n det x x j ȳ = x y j 1 x j p = xj p x j j = n. We denote by T 2 M = T 2 M \ {} ere : M T 2 M s te null secton of te projecton π 2. Let us consder te tangent bundle of te dfferentable manfold T 2 M T T 2 M τ 2 T 2 M ere τ 2 s te canoncal projecton and te vertcal dstrbuton V : u T 2 M V u T{ u T 2 M locally } generated by te vector felds u u y p u T 2 M. lnear mappng Te follong F T 2 M J : χ T 2 M χ T 2 M defned by: J = J x y u T 2 M = J y p = 2 ISSN: Issue 9 Volume 9 September 21

2 s a tangent structure on T 2 M. We denote t N a nonlnear connecton on te manfold T 2 M t te local coeffcents N j x y p N j x y p j = n. Hence te tangent space of T 2 M n te pont u T 2 M s gven by te drect sum of vector spaces: T u T 2 M = N u W 1 u W 2 u u T 2 M. 3 A local adapted bass to te drect decomposton 3 s gven by: { δ δx y p } = n 4 δ δx = N j x + N y j j p j. 5 Wt respect to te coordnates transformatons 1 e ave te rules: by: δ δx y p = xj δ ; x δ x j = xj ; x ȳ j = x x j p j. 5 Te dual bass of te adapted bass 4 s gven { δx δy δp } 6 δx = dx δy = dy + N jdx j 6 δp = dp N j dx j. Wt respect to 1 te covector felds 6 are transformed by te rules: δ x = x x j δx j δȳ = x x j δy j δ p = xj x δp j. 6 Let D be an N lnear connecton on T 2 M t te local coeffcents n te adapted bass: DΓ N = H C C. An N lnear connecton D s unquely represented n te adapted bass 4 n te follong form: δ D δ = H δx j δx j δ δx D δ = H δx j y j y D δ δx j p = H j p δ D = C y j δx j δ δx D = C y j y j y 7 D y j p = C j p δ = C j δ δx D p j D p j D p j δx y = C j y p = C j p. An N-lnear connecton D t te local coeffcents DΓ N = H C C determnes te 1 2 covarant dervatves n te tensor algebra of d-tensor felds. Defnton 1 [1] An N-lnear connecton on T 2 M s called semsymmetrc f: T = 1 2 δj σ + δ σ j S = 1 2 δj τ + δ τ j S = 1 2 δ j v + δ vj ere σ τ χ T 2 M and v χ T 2 M. 8 2 Conformal metrcal N-lnear connectons n a generalzed Hamlton space Defnton 2 [4] A generalzed Hamlton space of order to s a par GH 2n = M g j x y p 1 g j s a d tensor feld of type 2 symmetrc and nondegenerate on te manfold T 2 M. 2 Te quadratc form g j X X j as a constant sgnature on T 2 M. g j s called te fundamental tensor or metrc tensor of te space GH 2n. In te case en T 2 M s a paracompact manfold ten on T 2 M tere exst te metrc tensors g j x y p postvely defned suc tat M g j s a generalzed Hamlton space. Defnton 3 [4] A generalzed Hamlton metrc g j x y p of order to on sort GH metrc s ISSN: Issue 9 Volume 9 September 21

3 called reductble to an Hamlton metrc H metrc of order to f tere exsts a functon H x y p on T 2 M suc tat: 2 H g j = 1 2 p p j. 9 Te covarant tensor feld g j s obtaned from te equatons: g j g = δ. 1 g j s a symmetrc nondegenerate and covarant of order to d tensor feld. Defnton 4 [4] An N lnear connecton D s called metrcal t respect to GH metrc g j f: g j = g j = g j =. 11 Te tensoral equatons 11 mply: g j = g j = g j =. 12 Teorem 5 [4]1. Tere exsts a unque N lnear connecton D Γ N = H C C avng te propertes: 1. Te nonlnear connecton N s a pror gven. 2. D Γ N s metrcal t respect to GHmetrc g j.e.12 are verfed. S 3. Te torson tensors T S and vans. 2. Te prevous connecton as te coeffcents C and C gven by and C = 1 2 gm g m C = 1 2 g m y j g m p j + g jm y + gjm p g y m g p m H are generalzed Crstoffel symbols: 13 H = 1 2 gm δg m + δg jm δg δx j δx δx. 14 m Te operators of Obata s type are gven by: Ω j = 1 2 Ω j = 1 2 δ δj g g j δ δj + g g j. 15 Te operators of Obata s type ave te same propertes as te one assocated t a Fnsler space [3]. Let S 2 T 2 M be te set of all symmetrc d- tensor felds of te type 2. As s easly son te relatons for a j b j S 2 T 2 M defned by: a j b j λx y p FT 2 M a j x y p = e 2λxyp b j x y p s an equvalence relaton on S 2 T 2 M. 16 Defnton 6 Te equvalent class ĝ of S 2 T 2 M/ to c te fundamental d-tensor feld g j belongs s called conformal metrcal d-structure. Tus: ĝ = {g g j x y p = e2λxyp g j x y p λx y p FT 2 M}. 17 Defnton 7 An N-lnear connecton D t local coeffcents: DΓ N = H C C for c tere exsts te 1-form ω ω = ω dx + ω δy + ω δp suc tat: { gj = 2ω g j g j = 2 ω g j g j = 2 ω 18 g j ere and denote te 1 and 2 covarant dervatves t respect to D s called conformal metrcal N-lnear connecton t respect to te conformal metrcal d-structure ĝ correspondng to te 1-form ω and s denoted by: DΓN ω. Proposton 8 If DΓN ω = H C C are te local coeffcents of a conformal metrcal N- lnear connecton n T 2 M t respect to te conformal metrcal structure ĝ correspondng to te 1- form ω ten: { g j = 2ω g j g j = 2 ω g j g j = 2 ω g j 19. Proof. Usng te relatons 18 by covarant dervaton from 1 e ave te results. Proposton 9 Te operators of Obata s type are covarant constant t respect to any conformal metrcal N-lnear connecton D: { Ω r sj l = Ω r sj l = Ω r Ω r sj l = Ω r sj l = Ω r sj l = sj l = 2 ere l l and l denote te 1 and 2 covarant dervatves t respect to D. ISSN: Issue 9 Volume 9 September 21

4 Proof. Usng te relatons 18 and 19 by covarant dervaton from 15 e ave te results. For any representatve g ĝ e ave: Teorem 1 For g j = e 2λ g j a conformal metrcal N-lnear connecton t respect to te conformal metrcal structure ĝ correspondng to te 1-form ω DΓN ω satsfes: { g j = 2ω g j g j = 2 ω g j g j = 2 ω g j 21 ere ω = ω + dλ. Snce n Teorem 1 ω = s equvalent to ω = d λ e ave: Teorem 11 A conformal metrcal N-lnear connecton t respect to ĝ correspondng to te 1-form ω DΓN ω s metrcal t respect to g ĝ.e. g j = g j = g j = f and only f ω s exact. We sall determne te set of all conformal metrcal N-lnear connectons t respect to ĝ. Let D Γ N = H C C be te local coeffcents of a fxed N- lnear connecton D on T 2 M ere N j x y p N j x y p j = n are te local coeffcents of te nonlnear connecton N. Ten any N-lnear connecton D on T 2 M t te local coeffcents DΓN = H C C ere N j x y p N j x y p j = n are te local coeffcents of te nonlnear connecton N can be expressed n te form [6]: N j = N j A j N j = N j A j H = H + A l C jl A l C l j B 22 C = C D C = C D j = n t A = A = j = n j 23 j ere denotes te -covarant dervatve t respect to D and A j A j B D D are te components of te dfference tensor felds of D from D. Teorem 12 Let D be a gven N -lnear connecton t local coeffcents D Γ N = H C C. Te set of all conformal metrcal N-lnear connectons t respect to ĝ correspondng to te 1-form ω t local coeffcents DΓN ω = H C C s gven by: N j = N j X j N j = N j X j H = H + X l C jl X l C l j gm g 2ω g mj + g mj l X l mj g mj l X l + Ω r sj Xs r C = C gm g mj 2 ω g mj + +Ω r sj Y s r C = C g m 2 ω g m + +Ω rj j = n 24 t: s Z r s X = X j j = j = n 25 ere and denote te 1 and 2 covarant dervatves t respect to D X j X j X Y Z are arbtrary d-tensor felds ω = ω dx + ω δy + ω δp s an arbtrary 1-form and Ω s te operator of Obata s type gven by 15. Proof. Usng te relatons by extenson of te metod gven by R.Mron n [3] for te case of Fnsler connectons e obtan te results. Partcular cases: 1. If X j = X j = X = Y = Z = n Teorem 12 e ave: Teorem 13 Let D be a gven N -lnear connecton on T 2 M t local coeffcents D Γ N = H C C. Ten te follong N-lnear conecton K t local coeffcents KΓ N ω = ISSN: Issue 9 Volume 9 September 21

5 H C C gven by 26 s conformal metrcal t respect to ĝ correspondng to te 1-form ω: H = H gm g 2ω g mj mj C = C gm g mj 2 ω g mj C = C gjm g m 2 ω g m j = n 26 ere and denote te 1 and 2 covarant dervatves t respect to D and ω = ω dx + ω δy + ω δp s an arbtrary 1-form. 2. If e tae a metrcal N-lnear connecton as D n Teorem 13 ten 26 becomes: H = H δj ω C = C δj ω 27 C = C δ j ω j = n. As an exemple of D e tae te N-lnear connecton gven n Teorem Teorem 14 Te follong N-lnear connecton W t local coeffcents W ΓN ω = H C C s a conformal metrcal N- lnear connecton t respect to ĝ correspondng to te 1-form ω: H = 1 2 gm δg m + δg jm δg δx j δx δx m δj ω 2Ω m ω m C = 1 2 gm g m + g jm g y j y y m j ω 2Ω m ω m C = 1 2 g g m m p j + gjm p g p m j ω 2Ω m ωm j = n 28 ere ω = ω dx + ω δy + ω δp s an arbtrary 1-form. 4. If e tae a conformal metrcal N-lnear connecton t respect to ĝ e.g. W as D n Teorem 12 e ave: Teorem 15 Let D be a fxed conformal metrcal N- lnear connecton t respect to ĝ correspondng to te 1-form ω t te local coeffcents D Γ N ω = H C C. Te set of all conformal metrcal N-lnear connectons t respect to ĝ correspondng to te 1-form ω t local coeffcents DΓN ω = H C C s gven by: N j = N j X j N j = N j X j H = H + C jl + ω l δj X l C l j + ω l δj X l + Ω r sj Xs r 29 t C = C + Ω r sj Y s r C = C + Ω jr j = n X = X j j s Z r s = j = n 3 ere and denote te 1 and 2 covarant dervatves t respect to D ω = ω dx + ω δy + ω δp s an arbtrary 1-form and X j X j X Y Z are arbtrary d-tensor felds. 5. Fnally f e tae X j = X j = n Teorem 15 e obtan: Teorem 16 Let D be a fxed conformal metrcal N- lnear connecton t respect to ĝ correspondng to te 1-form ω t local coeffcents D Γ N ω = H C C. Te set of all conformal met- rcal N-lnear connectons t respect to ĝ correspondng to te 1-form ω correspondng to te same nonlnear connecton N t local coeffcents DΓ N ω = H C C s gven by: H = H + Ω r sj Xs r C = C + Ω r sj Y s r 31 C = C + Ω jr j = n s Z r s ISSN: Issue 9 Volume 9 September 21

6 ere X Y Z are arbtrary d-tensor felds on T 2 M and ω = ω dx + ω δy + ω δp s an arbtrary 1-form. 3 Some specal classes of conformal metrcal N-lnear connectons We sall try to replace te arbtrary tensor felds X Y and Z n Teorem 16 by te torson tensor felds T S and S. We put: T = 1 2 gm g m T g jt m + +g T jm S = 1 2 gm g m S g js m + +g S jm S = 1 2 g mg m S +g S jm. gj S m + 32 Teorem 17 Let T S and S be tree gven se symmetrc tensor felds of type and 21 respectvely and let ω be a gven 1-form n T 2 M. Ten tere exsts a unque conformal metrcal N-lnear connecton t respect to ĝ correspondng to te 1-form ω t local coeffcents DΓN ω = H C C avng T S and S as te torson tensor felds. It s gven by: H = H + T C = C + S C = C + S 33 ere W ΓN ω = H C C are te local coeffcents of conformal metrcal N-lnear connecton t respect to ĝ correspondng to te 1-form ω gven n 28. Remar 18 Te conformal metrcal N-lnear connecton t respect to ĝ W correspondng to te 1-form ω t local coeffcents W ΓN ω = H C C gven n 28 s consdered as te semsymmetrc conformal metrcal N-lnear connecton t te vansng 1 and 2 torson vector felds. Usng te Defnton 1 te relatons 32 become: T = 2Ωr σ r S = 2Ωr τ r 34 S = 2Ω r vr. Usng te Teorem 17 and te relatons 34 e ave: Teorem 19 Te set of all semsymmetrc conformal metrcal N-lnear connectons t respect to ĝ correspondng to te 1-form ω t local coeffcents DΓN ω σ = H C C s gven by: H = H + 2Ω r σ r C = C + 2Ω r τ r 35 C = C + 2Ω r vr j = n ere W ΓN ω = H C C are te local coeffcents of te semsymmetrc conformal metrcal N-lnear connecton W gven n 28 and σ = σ dx + τ δy + v δp s an arbtrary 1-form. 4 Te group of transformatons of conformal metrcal N-lnear connectons We study te transformatons DΓN ω DΓ N ω of te conformal metrcal N-lnear connectons t respect to ĝ. If e replace D Γ N and DΓN ω n Teorem 12 by DΓN ω and DΓ N ω respectvely to conformal metrcal N- and respectvely N-lnear connectons t respect to ĝ e obtan: Teorem 2 To conformal metrcal N- and respectvely N- lnear connectons t respect to ĝ: D and D t local coeffcents DΓN ω = H C C and DΓ N ω = H C related as follos: N j = N j X j N j = N j X j C H = H + Xl C jl X lcj l δj p + δ j ω lx l δ j ωl X l + Ω r C = C δj ṗ + Ωr sj Y s r C = C δ j p + Ω jr s Z s j = n r respectvely are sj Xs r 36 ISSN: Issue 9 Volume 9 September 21

7 t: X j = X j = j = n 37 ere p = ω ω ω = ω dx + ω δy + ω δp and ω = ω dx + ω δy + ω δp are to 1-forms denote te -covarant dervatve t respect to D and X j X j X Y Z are arbtrary d-tensor felds. Proof. Usng n 24 te relatons 18 by drect calculaton e ave te results. Conversely gven te d-tensor felds X j X j X Y Z and one gven 1-form p = p dx + ṗ δy + p δp te above 36 s tougt to be a transformaton of a conformal metrcal N- lnear connecton DΓN ω to a conformal metrcal N-lnear connecton DΓ N ω = DΓ N ω + p. We sall denote ts transformaton by tx j X j X Y Z p. Tus e ave: Teorem 21 Te set C of all transformatons tx j X j X Y Z p gven by 36 and 37 s a transformatons group of te set of all conformal metrcal N-lnear connectons t respect to ĝ togeter t te mappng product: tx j X j X Y Z p tx j X j X Y Z p = tx j + X j X j + X j X + X Y + Y Z + Z p + p. 5 Te group of transformatons of semsymmetrc conformal metrcal N-lnear connectons We nqure about a subgroup of te group of transformatons of conformal metrcal n-lnear connecton: about te subgroup of transformatons of te semsymmetrc comformal metrcal N-lnear connectons correspondng to te same nonlnear connecton N. Let N be a gven nonlnear connecton. Ten any semsymmetrc conformal metrcal N-lnear connecton t local coeffcents DΓN ω σ = H C C t respect to ĝ s gven by 33 t 34. Teorem 22 To semsymmetrc conformal metrcal N-lnear connectons t respect to ĝ t local coeffcents DΓN ω σ = H C C and DΓ N ω σ = H C are related as follos: C H = H δ j p + 2Ωr q r C = C δj ṗ + 2Ωr q r C = C j = n δ j p + 2Ω r qr respectvely 38 ere p = ω ω q = σ σ p p = p dx + ṗ δy + p δp and q = q dx + q δy + q δp. Proof. Usng n 35 te relatons 28 by drect calculaton e ave te results. Conversely gven 1-forms p and q n T 2 M te above 38 s tougt to be a transformaton of a semsymmetrc conformal metrcal N-lnear connecton D t local coeffcents DΓN ω σ = H C C to a semsymmetrc conformal metrcal N- lnear connecton D t local coeffcents DΓN ω + p σ + p + q = H C C. We sall denote ts transformaton by tp q. Tus e ave: Teorem 23 Te set CN s of all transformatons tp q gven by 38 s a transformatons group of te set of all semsymmetrc conformal metrcal N-lnear connectons t respect to ĝ avng te same nonlnear connecton N togeter t te mappng product: tp q tp q = tp + p q + q. Ts group CN s s an Abelan subgroup of C and acts on te set of all semsymmetrc conformal metrcal N-lnear connectons avng te same nonlnear connecton N transtvely. Te transformaton tp q : DΓN ω σ DΓN ω + p σ + p + q gven by 38 s expressed by te product of te follong to transformatons: H = H δ j p C = C δ jṗ C = C δ j 39 p j = n H = H + 2Ωr q r C = C + 2Ωr q r C = C + 2Ω 4 r qr j = n Defnton 24 Te transformaton t : DΓN DΓN of N-lnear connecton on T 2 M defned by ISSN: Issue 9 Volume 9 September 21

8 39 s called co-parallel transformaton ere p s a gven 1-form. Teorem 25 Te set C p N of all co-parallel transformatons t gven by 39 s an Abelan group togeter t te mappng product. Defnton 26 Te transformaton t : DΓN DΓN of N-lnear connectons gven by 4 s called Mron transformaton as te name gven by M.Hasguc [3] for Fnsler spaces. Teorem 27 Te set CN m of all Mron transformatons t gven by 4 s a transformatons group togeter t te mappng product. Teorem 28 Te group CN s of all transformatons tp q gven by 38 s te drect product of te group C p N of all co-paralel transformatons and te group of all Mron transformatons. C m N It s noted tat te nvarants of te group CN s ll be te nvarants of eac of tese subgroups and recprocally. It s drectly son tat by a co-parallel transformaton 39 te curvature tensor felds R P and S are transformed as follos: R = R δ p P = P δ ṗ S = S δ p 41 ere p ṗ and p are te components of dp expressed t respect to D. Elmnatng p ṗ and p from 41 e ave: R = R S Tus e ave: P = P = S R = R 1 n δ R s s s s. P = P 1 n δ P s S = S 1 n δ S s Teorem 29 Te tensor felds R P and S gven by 43 are nvarants of te group C p N. Also e obtan: Teorem 3 Te tensor feld C gven by 44 s an nvarant of te group C p N. C = C 1 n δj C s s. 44 In our prevous paper [Bull Mat Buc] startng from te tensor felds: K = R C m Rm 1 C m R 2m P = P C m N jm p C m N m j y 45 e obtaned te follong mportant nvarants of te group of semsymmetrc metrcal N-lnear connectons avng te same nonlnear connecton N T ms N for n > 2: H = K + 2 n 2 A {Ω r j K r g rk 2n 1 } N = P + 2 n 2 A {Ω r j P r g rp 2n 1 } M = S + 2 n 2 A {Ω r j S r g rs M 2n 1 } = S + 2 n 2 A {Ω j r Sr grj S 2n 1 } K j = K j K = gj K j P j = P P = g j P j S j = S j S = gj S j S j = Sm jm S = g j S j j If e replace tese K P S j and S by te tensor felds K P S and S respectvely defned by: K = K 1 n δ K m m m m m P = P 1 n δ P m S = S 1 n δ S m S = S 1 n δ S m 48 e can obtan te nvarants of te group of transformatons of semsymmetrc conformal metrcal N- lnear connectons avng te same nonlnear connecton N C s N : Teorem 31 For n > 2 te follong tensor felds H N M and M are nvarants of te group CN s of transformatons of semsymmetrc conformal metrcal N-lnear connectons avng te same nonlnear connecton N: ISSN: Issue 9 Volume 9 September 21

9 H = K + 2 n 2 A {Ω r j K r g rk 2n 1 } N = P + 2 n 2 A {Ω r j P r g rp 2n 1 } M = S + 2 n 2 A {Ω r M g rs 2n 1 } = S + 2 n 2 A {Ω j r S r gr S 2n 1 } j S r K j = K j K = g j K j P j = P j P = g j P j S j = S j S = g j S j S j = S jm m S = g j S j. Fnally e gve anoter nvarant of te group C s N : 49 5 Teorem 32 Te follong tensor feld s an nvarant of te group C s N : C 2 n 1 Ωj r j = n ere C s gven by 44. C rm m 51 6 Metrcal N-lnear connectons n a generalzed Hamlton space We sall determne te set of all metrcal N-lnear connectons n te case en te nonlnear connecton N s arbtrary. Teorem 33 Let D be gven N-lnear connecton on T 2 M t local coeffcents D Γ N = = H C C ere te local coeffcents of te nonlnear connecton N are: N j x y p N j x y p j = n. Te set of all metrcal N-lnear connectons t respect to g j t local coeffcents DΓN = H C C s gven by: N j = N j X j N j = N j X j H = H + X l t: C jl X l C j l gm g + g mj l X l mj g mj l X l + Ω r sj Xs r C = C gm g mj + Ω r sj Y s r C = C gmj g m + Ω rj s Z r s j = n 52 X = X = j = n 53 j j ere and denote te 1 and 2 covarant dervatves t respect to D X j X j X Y Z are arbtrary d-tensor felds and Ω s te operator of Obata s type gven by 15. Proof. Usng te relatons by extenson of te metod gven by R.Mron n [3] for te case of Fnsler connectons e can deduce te results. Partcular cases: 1. If X j = X j = X = Y = Z = n Teorem 33 e ave: Teorem 34 Let D be a gven N-lnear connecton on T 2 M t local coeffcents D Γ N = H C C. Ten te follong N-lnear connecton D t local coeffcents DΓ N = H C C gven by 54 s metrcal: N j = N j N j = N j H = H gm g mj C = C gm g mj C = C gmj g m j = n 54 ISSN: Issue 9 Volume 9 September 21

10 ere and denote te 1 and 2 covarant dervatves t respect to D. 2. If e tae a metrcal N-lnear connecton as D n Teorem 33 e obtan: Teorem 35 Te set of all metrcal N- lnear connectons t local coeffcents DΓN = H C C s gven by: N j = N j X j N j = N j X j H = t: H + X l C jl X l C l j + Ω r sj Xs r C = C + Ω r C = C + Ω rj j = n X = X j ere j and sj Y s r s Z r s 55 = j = n 56 denote te 1 and 2 covarant dervatves t respect to D X j X j X Y Z are arbtrary d-tensor felds and Ω s te operator of Obata s type gven by If n Teorem 35 e consder X = X j = o e obtan te set of all metrcal N-lnear connectons avng te same nonlnear connecton N gven by R.Mron D.Hrmuc H.Smada and V.S. Sabău n ter boo [4] Teorem 2.3 p Te group of transformatons of metrcal N-lnear connectons Let N be a gven nonlnear connecton. Ten any metrcal N-lnear connecton correspondng to te same nonlnear connecton N as te local coeffcents DΓN = H C C. gven by: H = H + Ωr sj Xs r C = C + Ωr sj Y s r C = C + Ω rj s Zr s j = n 57 ere X Y Z are arbtrary d-tensor felds Ω s te operator of Obata s type gven by 15 and DΓN = H C C are te local coeffcents of a metrcal N-lnear connecton D. Conversely gven te tensor felds X Y Z te above 57 s tougt to be a transformaton of a metrcal N-lnear connecton DΓN to a metrcal N-lnear connecton DΓN. We sall denote ts transformaton by tx Y Z. Tus e ave: m T N Teorem 36 Te set of all transformatons tx Y Z gven by 57 togeter t te mappng product: tx Y Z tx = tx + X Y + Y Z Y Z Z + s a transformatons group of te set of all metrcal N-lnear connectons avng te same nonlnear connecton N. References: [1] G. Atanasu and M. Târnoveanu Ne Aspects n te Dfferental Geometry of te Second Order Cotangent BundleUnv. Tmşoara No [2] R. Mron Hamlton Geometry Semnarul de Mecancă Unv. Tmşoara [3] R. Mron and M. Hasguc Conformal Fnsler Connectons Rev.Roumane Mat.Pures Appl [4] R. Mron D. Hrmuc H. Smada and V. S. Sabău Te Geometry of Hamlton and Lagrange Spaces Kluer Acad.Publ. Vol 118FTPH 21. [5] R. Mron S. Watanabe and S. Ieda Cotangent Bundle Geometry Memorle Secţlor ştnţfce Bucureşt Acad. R.S.Romana Sera IVIX I [6] M. Purcaru and M. Târnoveanu On Transformatons Groups of N-Lnear Connectons on Second Order Cotangent Bundle Acta Unverstats Apulenss Specal Issue [7] M. Purcaru and M. Târnoveanu Metrcal Semsymmetrc N-Lnear Connectons on a Generalzed Hamlton Space Bull. Mat. Soc. Sc. Mat. Roum to appear. [8] C. Udrşte Mult-Tme Optmal Control WSEAS Transactons on Matematcs Issue 12 Volume 6 December 27 ISSN: ISSN: Issue 9 Volume 9 September 21

11 [9] C. Udrşte Non-classcal Lagrangan dynamcs and potental maps WSEAS Transactons on Matematcs Volume ISSN: [1] C. Udrşte and D. Oprş Euler-Lagrange- Hamlton dynamcs t fractonal acton WSEAS Transactons on Matematcs Volume ISSN: [11] C. Udrşte M. Popescu and P. Popescu Generalzed Multtme Lagrangans and Hamltonans WSEAS Transactons on Matematcs Volume Scopus Scmago ISSN: [12] C. Udrşte and O. Şandru Dual Nonlnear Connectons communcate to te 22 nd Conference on Dfferental Geometry and Topology Polytecnc Insttute of Bucarest Romana Sept [13] C. Udrste I. Tevy Mult-Tme Euler-Lagrange- Hamlton Teory WSEAS Transactons on Matematcs Volume ISSN: [14] K. Yano and S. Isara Tangent and Cotangent Bundles. Dfferental Geometry M.Deer Inc. Ne-Yor ISSN: Issue 9 Volume 9 September 21