On Integrability Conditions of Derivation Equations in a Subspace of Asymmetric Affine Connection Space

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1 Flomat 9:0 (05), 4 47 DOI 0.98/FI504Z ublshed by Faculty of Scences and Mathematcs, Unversty of Nš, Serba valable at: htt:// On Integrablty Condtons of Dervaton Equatons n a Subsace of symmetrc ffne Connecton Sace Mlan j. Zlatanovć a, Svetslav M. Mnčć a, jubca S. Velmrovć a a Faculty of Scence and Mathematcs, Unversty of Nš, Serba bstract.in a sace N of asymmetrc affne connecton by equatons (.) a submanfold X M N s defned. On X M and on seudonormal submanfold X N asymmetrc nduced connectons are defned. N M Because of asymmetry of nduced connecton t s ossble to defne four knds of covarant dervatve. In ths work we are consderng ntegrablty condtons of dervatonal equatons [4] obtaned by hel of the st and the nd knd of covarant dervatve. The corresondng Gauss-Codazz equatons are obtaned too.. Introducton Consder a sace N of asymmetrc affne connecton wth a torson (n local coordnates) T jk = jk kj. Saces wth asymmetrc affne connecton and ther roertes were studed by many authors [, ]. submanfold X M N s defned by equatons x = x (u,..., u M ) = x (u ), =, N. (.) artal dervatves B = x u (rank(b ) = M) defne tangent vectors on X M. Consder N M contravarant vectors C (, B, C,..., {M +,..., N}) defned on X M and lnearly ndeendent, and let the matrx ( B ) C be nverse for the matrx (B, C ) rovded that the followng condtons are satsfed [0]: a) B B β = δ β ; b) B C = 0; c) B C = 0; d) C CB = δ B ; e) B B j + C C j = δ j ; (.) The magntudes B, B are rojecton factors (tangent vectors), and the magntudes C, C are affne seudonormals [4, 0]. 00 Mathematcs Subject Classfcaton. rmary 5B05; Secondary 5C5, 545 Keywords. Dervaton equatons, submanfold, asymmetrc connecton, nduced connecton, ntegrablty condtons eceved: February 05; cceted: 4 rl 05 Communcated by Mća Stankovć esearch suorted by MNT grant 740 Emal addresses: zlatmlan@mf.n.ac.rs (Mlan j. Zlatanovć), svetslavmncc@yahoo.com (Svetslav M. Mnčć), vljubca@mf.n.ac.rs (jubca S. Velmrovć)

2 The nduced connecton on X M s [4, 0, ]: M.j. Zlatanovć al. / Flomat 9:0 (05), βγ = B (B β,γ + jk Bj β Bk γ), (.) where B β,γ = B β / uγ = x / u β u γ. Because s asymmetrc by vrtue of j, k, s asymmetrc n β, γ too. The submanfold X M endowed wth becomes M and we wrte M N. The set of seudonormals of the submanfold X M N makes a seudonormal bundle of X M, and we note t X N. We have defned n [4] nduced connecton of seudonormal bundle wth coeffcents N M Bµ = C (C B,µ + jkc j B Bk µ). kj (.4) s the coeffcents,, are generally asymmetrc, we can defne four knds of covarant dervatve for a tensor, defned n the onts of X M. For examle: t jβb µ = t jβb,µ + mt jβb t jm m mj m mj 4 m jm βb + µ µ µ µ t jβb βµ µβ µβ βµ t jb + µ t jβb βµ t jβ, (.5) In ths manner four connectons on X M N are defned. We shall note the obtaned structures (X M N,, {,..., 4}).. Integrablty condtons of dervatonal equatons for tangents.0 From (.7, 7 ) n [4], we have a dervatonal equatons for tangents B µ = Ω µc, B µ = Ω µ C and from (.9, 9 ) for seudonormals C µ = Ω µ B, C µ = Ω So, one obtans, {, }, (.) µb B µ, {, }. (.) ν = (Ω = (.) Ω µc ) ν = Ω µ µ Ω ν B + Ω ν C + Ω µ ν C, where = (.) sgnfes equal wth resect to (.). In ths manner B µ ν B ν µ = ( Ω µ Ω and by analogous rocedure: ν Ω µ)b + (Ω µ ν Ω ν µc ν µ )C, (.) where [4] B µ ν B ν µ = (Ω µ Ω ν Ω ν Ω µ )B + ( Ω µ ν µ )C, (. ) Ω µ = C (B,µ + mb B m µ ), m (.4)

3 where Based on the cc tye denttes [, ] B µ ν B ν µ = M.j. Zlatanovć al. / Flomat 9:0 (05), Ω µ = C (B,µ m m B B m µ ). (.4 ) mn B B m µ B n ν µν B + ( ) T µνb, {, }. (.5) jmn = jm,n jn,m + jm n jn m, (.6) jmn = mj,n nj,m + mj n nj m, (.7) are curvature tensors of the st resectvely the nd knd of the N and, analogously, βµν, are curvature βµν tensors of M N, obtaned n the same manner by connecton βγ.. Takng (.) = {, }, equalzng the rght sdes of obtaned equaton and (.5) and exchange B wth resect of (.), we get the st and the nd knd ntegrablty condton of dervatonal equaton (.): mn B B m µ B n ν = ( µ + (Ω µ ν Ω µ Ω ν ν Ω µ)b µ + ( ) T µνω )C, {, }. (.8) a) Comosng that equaton wth B and takng nto consderaton (.), we obtan µν = mn B B B m µ B n ν + Ω µ Ω ν Ω ν Ω µ, {, }. (.9) whch are Gauss equaton of the st and the nd knd ( =, ) for M N. b) If one comoses (.8) wth C, we get mn C B B m µ B n ν = Ω µ ν Ω ν µ + ( ) T µνω, {, }. (.0) and that are the st Codazz equaton of the st and the nd knd ( =, ) for M N.. Startng from the cc tye denttes (. ) for {, } B µ ν B ν µ = mn B B m µ B n ν + µν B + ( ) T µνb, {, }. (.5 ) substtutng B nto (.5 ) by vrtue of (. ), we get the st and the nd ntegrablty condton of dervaton equaton (. ): mn B B m µ B n ν = ( µν Ω µ Ω ν + Ω ν Ω + ( Ω µ ν a ) By comosng the revous equaton wth B one gets µ )B µ + ( ) T µν Ω )C, {, }. (.8 ) µν = mn B B Bm µ B n ν + Ω µ Ω ν Ω ν Ω µ, {, },

4 M.j. Zlatanovć al. / Flomat 9:0 (05), and that equaton by exchanges, becomes (.9). b ) Comosng (.8 ) wth C, t follows that mn C B B m µ B n ν = Ω µ ν µ + ( ) T µν Ω, {, }, (.0 ) whch s the another form of the st Codazz equaton of the st and the nd knd for M N.. By alcaton of the corresondng cc tye dentty [, ], one obtans where B µ ν B ν µ = µν B µν B, (.) jµν = ( jm,n nj,m + jm n nj m)b m µ B n ν + T jm (Bm µ,ν νµb m ), (.) s the rd knd curvature tensor of N relatng to M, and βµν = βµ,ν νβ,µ + βµ ν νβ µ + νµ T β (.) s the rd knd curvature tensor of the subsace M N. On the other hand, uttng at (.) =, =, and comarng the obtaned equaton wth (.), we have the rd ntegrablty condton of dervatonal equaton (.). µν B µν B = ( Ω µ Ω ν Ω a) By comosng wth B, from here s obtaned µ)b + (Ω µ ν Ω ν µ )C. (.4) µν = µν B B + Ω µ Ω ν Ω ν Ω µ, (.5) whch s Gauss equaton of the rd knd for M N. b) Comosng (.4) wth C, one gets µν C B = Ω µ ν Ω µ ν, (.6) and ths s st Codazz equaton of the rd knd for M N.. For B, usng the cc tye dentty [, ], t s B µ ν B ν µ = µν B + µν B, (. ) uttng nto (. ) =, =, by comarng the obtaned equaton and (. ), we obtan the rd ntegrablty condton of dervatonal equaton (. ). µν B + µν B Therefrom, analogously to revous case, we have = (Ω µ Ω ν Ω ν Ω µ )B + ( Ω µν = µν B B + Ω µ Ω ν Ω ν Ω µ µ. ν µ )C. (.4 )

5 M.j. Zlatanovć al. / Flomat 9:0 (05), Dong an exchange of ndces,, we see that ths equaton reduce to (.5). By comosng (.4 ) wth C, one obtans µν C B = Ω µ and ths s another form of the st Codazz equaton of the rd knd for M N. Based on exosed, the followng theorems are vald. µ ν, (.6 ) Theorem.. The st and nd knd ntegrablty condtons of dervatonal equatons (.) and (. ) for submanfold X M N wth the structure (X M N,, {, }), where the connectons are defned n (.5), are gven n (.8) and (.8 ) resectvely. The rd knd ntegrablty condtons for the mentoned equatons are (.4) and (.4 ). Theorem.. Gauss equatons of the st and the nd knd are gven n (.9), and of the rd one n (.5). The st Codazz equatons of the st and the nd knd are gven n (.0), and of the rd knd n (.6). The equatons (.0 ), (.6 ) are another forms of (.0) and (.6) resectvely.. Integrablty condtons of dervatonal equatons for seudonormals.0 In order to obtan ntegrablty condtons for dervatonal equatons of seudonormals, we treat analogously to the case of tangents. From (.,) one obtans C µ ν C ν and from (., ): µ = ( Ω µ C µ ν C ν ν µ )B ( Ω µ = (Ω µ µ Ω ν Ω By vrtue of the cc tye dentty from [] s C µ ν C ν ν ν Ω µ)c (.) µ )B (Ω µ Ω ν Ω ν Ω µ )C. (. ) µ = mn C Bm µ B n ν µν C + ( ) T µνc, {, }, (.) and also one can rove based on (.5) that where [] C µ ν C ν µ = mn C B m µ B n ν + µν C + ( ) T µνc, {, }, (. ) Bµν = Bµ,ν Bν,µ + Bµ ν Bµ ν, {, } (.) are the st and the nd knd curvature tensors of N wth resect to the seudonormal submanfold X N N M.. If one substtutes = {, } nto (.) and equalzes the rght sdes of obtaned equaton and (.), exchangng revously C wth hel of (.), one obtans mn C Bm µ B n ν = [( ) T σ µν Ω σ Ω + ( µ µ Ω µ ν Ω ν µ ]B µ)c, {, }, (.4) and ths are the st and nd knd ntegrablty condtons for seudonormals of dervatonal equaton (.).

6 M.j. Zlatanovć al. / Flomat 9:0 (05), a) If one comoses (.4) wth B, t s obtaned that by vrtue of (.): mn B C Bm µ B n ν = ( ) T σ µν Ω σ Ω Exchangng here,,, σ, we obtan (.0 ). b) Comosng (.4) wth C and usng (.) t follows that mn C C Bm µ B n ν = µ µ Ω ν Ω µ ν µ. µ, {, }. (.5) The equaton (.5) s the nd Codazz equaton of the st and the nd knd for X M N.. Takng = {, } n (. ), substtutng n (. ) the corresondng value of C and equalzng the rght sdes of mentoned equatons we have mn C B m µ B n ν = [ ( ) T σ µνω σ + Ω µ ν ν Ω µ ]B (.4 ) + ( µν + Ω µ Ω ν Ω ν Ω µ )C, {, }, whch s the st and the nd knd ntegrablty condtons ( =, ) of dervatonal equaton (. ). a) Comosng the revous equaton wth B we get mn C B Bm µ B n ν = ( ) T σ µνω σ + Ω and that s, exchangng some ndces, the equaton (.0). b) If one comoses (.4 ) wth C, t follows that and that s (.5).. Wth resect of (.9) n [] s C µ ν C ν µ = µ ν Ω mn C C Bm µ B n ν = µν + Ω µ Ω ν Ω ν Ω ν µ µν C µν C, (.6) where s gven n (.) and ([], the equaton (.0)): µ, Bµν = Bµ,ν Bν,µ + Bµ ν Bν µ + νµ( B B ) (.7) s the thrd knd curvature tensor of the sace N wth resect of X N N M. On the other hand, uttng at (.) =, =, and equalzng the rght sdes of the obtaned equaton and (.6), we get µν C µν C = ( Ω µ ν µ )B ( Ω µ Ω ν Ω µ)c, (.8) and that s the rd knd ntegrablty condton for seudonormals of dervatonal equaton (.). a) By comosng (.8) wth B, we get µν C B = ( Ω µ ν µ ), (.9)

7 and by substtuton,, one obtans (.6 ). b) Comosng (.8) wth C, we have µν = µν C C + Ω M.j. Zlatanovć al. / Flomat 9:0 (05), µ Ω ν Ω µ (.0) what, comarng wth (.5) we call the nd Codazz equaton of the rd knd.. nalogcally to (.6), can be roved that the cc-tye dentty C µ ν C ν µ = µν C µν C, (.6 ) s vald, where, are gven n (.7) and (.) resectvely. If one takes =, = n (. ) and comares the obtaned equatons wth (.6 ), one obtans µν C µν C = (Ω µ ν Ω ν whch s rd knd ntegrablty condton of dervatonal equaton (. ). s n revous cases, from (.8 ) one gets µ )B + (Ω µ Ω ν Ω ν Ω µ )C (.8 ) Further, from (.8 ), we obtan (.0). µν C B = Ω µ ν Ω ν µ, Based on exosed n ths secton, we have followng theorems: Theorem.. The st and the nd knd ntegrablty condtons of dervatonal equatons (.), and (. ) for seudonormals of a submanfold X M N wth structure (X M N,, {, }), where the connectons are defned n (.5), are gven n (.4) and (.4 ) resectvely. The rd knd ntegrablty condtons for these equatons are (.8), (.8 ). Theorem.. The nd Codazz equaton of the st and the nd knd s gven by (.5), and of the rd one by (.0). eferences [] S. M. Mnčć, cc tye denttes n a subsace of a sace of non-summetrc affne connexon, ubl. Inst. Math. (Beograd), N.S., 8(), (975), [] S. M. Mnčć, j. S. Velmrovć, M. S. Stankovć, Integrablty condtons of dervatonal equatons of a submnfold n a generalzed emannan sace, led Mathematcs and Comutaton, 0(04) 6:9. [] S. M. Mnčć, j. S. Velmrovć, M. S. Stankovć, New ntegrablty condtons of dervatonal equatons of a submnfold n a generalzed emannan sace, Flomat 4: (00), [4] S. M. Mnčć, Dervatonal equatons of submanfolds n an asymmetrc affne connecton sace, Kragujevac Journal of Math., Vol. 5, No. (0), [5] M. S. Stankovć, Frst tye almost geodesc mangs of general affne connecton saces, Nov Sad J. Math. 9, No. (999), -. [6] M. S. Stankovć, On a canonc almost geodesc mangs of the second tye of affne saces, Flomat, (999), [7] M. S. Stankovć, On a secal almost geodesc mangs of the thrd tye of affne saces, Nov Sad J. Math., No. (00), 5-5. [8] M. S. Stankovć, Secal equtorson almost geodesc mangs of the thrd tye of non-symmetrc affne connecton saces, led Mathematcs and Comutaton, 44, (04), [9] j. S. Velmrovć, M. S. Ćrć, N. M. Velmrovć, On the Wllmore energy of shells under nfntesmal deformatons, Comuters & Mathematcs wth lcatons 6(), (0), [0] K., Yano, Sur la théore des deformatons nfntesmales, Journal of Fac. of Sc. Unv. of Tokyo, 6, (949), 75. [] K., Yano, Notes on nfntesmal varatons of submanfolds, J. Math. Soc. Jaan, Vol., No. 4, (980), 45 5.