ON RICCI TYPE IDENTITIES IN MANIFOLDS WITH NON-SYMMETRIC AFFINE CONNECTION. Svetislav M. Minčić

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1 PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle séie, tome 9 (08) (0), 0 7 DOI: 0.98/PIM080M ON RICCI TPE IDENTITIES IN MANIFOLDS WITH NON-SMMETRIC AFFINE CONNECTION Svetislav M. Minčić Abstact. In [8], using polylinea mappings, we obtained seveal cuvatue tensos in the space L N with non-symmetic affine connection. By the same method, we hee examine Ricci type identities.. Intoduction Conside N-dimensional diffeentiable manifold M N on which a non-symmetic affine connection is defined. If X(M N ) is a Lie algeba of smooth vecto fields and X, X(M N ), then the mapping : X(M N ) X(M N ) X(M N ) given by (.) X = X + [X, ] defines an othe non-symmetic connection on M N []. That means that we have + X = X + X, (X + X ) = X + X, f X = f X, (fx) = f X + f X, fo =, and X,, X, X,, X(M N ), f F(M N ), whee F(M N ) is an algeba of smooth eal functions on M N. In that case we wite L N = (M N,, ) and L N call a space on non-symmetic connections,. If we intoduce local coodinates x,..., x N and put / x i = i, in view of (.) it will be (.) j k = k j. 00 Mathematics Subject Classification: Pimay C0; Seconday B0. Key wods and phases: non-symmetic connection, cuvatue tensos, independent cuvatue tensos, Ricci type identities. 0

2 06 MINČIĆ Denoting coefficients of the connection in the base,..., N with L i jk, we have k j = L i jk i, k j = j k = L i kj i, whee = denotes equal with espect (.) (.) to (.)". Futhe, if we take by definition it follows T(X, ) = X X + [X, ], {, }, T(X, ) = T(X, ) T (X, ), ( T(X, ) = T(X, )) ( = = ). We poved in [8] how it is possible to obtain seveal cuvatue tensos in L N by polylinea mappings. It is poved that among these tensos thee ae independent ones: (.) (.) (.) R(X;, ) = X X + [,] X, R(X;, ) = X X + [,] X, R(X;, ) = X X + X X, (.6) R(X;, ) = X X + X X, (.7) R(X;, )= ( X X+ X X+ [,] X+ [,] X), while the est can be expessed as linea combinations of these five tensos. Fo X = / x j j, = k, = l, one obtains (.8) R i jkl = L i jk,l L i jl,k+l p jk Li pl L p jl Li pk, (.9) R i jkl = Li kj,l Li lj,k +Lp kj Li lp Lp lj Li kp, (.0) R i jkl = Li jk,l Li lj,k +Lp jk Li lp Lp lj Li pk +Lp lk (Li pj Li jp ), (.) R i jkl = L i jk,l L i lj,k+l p jk Li lp L p lj Li pk+l p kl (Li pj L i jp), (.) R i jkl = (Li jk,l +Li kj,l Li jl,k Li lj,k +Lp jk Li pl +Lp kj Li lk Lp jl Li kp Lp lj Li pk ). Identities fo a vecto and fo a covecto by both connections.. Conside an expession (.) ( X X)(ω),, {, }.

3 ON RICCI TPE IDENTITIES Let us quote the elations (fo =, ) ( X)(ω) = [X(ω)] X( ω), ( ω)(x) = [ω(x)] ω( X). and denote (.) a) X = X X(M N ), b) ω = ω X (M N ). Then we have ( X)(ω) = (a) ( X)(ω) = () [X(ω)] X( ω) (.) and one gets = [( X)(ω)] ( X)(ω) () = { [X(ω)] X( X)} { [X( ω)] X( ω)} (b) = [X(ω)] [X( ω)] [X( ω)] + X( ω), (.) ( X X)(ω) = [, ][X(ω)] X( ω ω), i.e., ( X X)(ω)=[, ][X(ω)] ( ω ω)(x),, {, }. Fom (.) ( [, ] X)(ω) = [, ] [X(ω)] X( [, ] ω) = [, ][X(ω)]+( [,] ω)(x), we find the fist addend on the ight side and substitute in (.7). We obtain (.6) ( X X + [,] X)(ω) = ( ω ω + [,] ω)(x),, {, } Definition.. The equations (.) fo, {, } ae Ricci type identities fo a vecto in L N. (.7).. Taking = =, we obtain the coesponding identity fo : Denoting (.8) R(X;, )(ω) = ( ω ω + [,] ω)(x). the equation (.7) gives a elation (.9) R(ω;, ) = ω ω + [,] ω, R(X;, )(ω) = R(ω;, )(X).

4 08 MINČIĆ In ode to wite the equation (.) in local coodinates fo = =, we take X = X j j, = k, = l, ω = dx i. Fo the left side in (.) we obtain L = ( l k X k l X)(dx i ) = [ l (X j k j) k (X j l j)](dx i ) = [(X j k ),l j + X j k Lp jl p (X j l ),k j X j l Lp jk p](dx i ) = (X j k ),lδj i + X j k Lp jl δi p (X j l ),kδj i X j = (X i k ),l + X j k Li jl (Xi l ),k X j = X i = X i k l + Lp kl Xi p Xi l k Lp lk Xi p k l Xi k l + T p kl Xi p. Fo the ight-hand side in (.) we obtain and fom L = R, we have (.0) X i l Li jk l Lp jk δi p R = [ l, k ](X(dx i )) + X( l k dx i k l dx i ) = 0 + X[ l ( L i pl dxp ) l ( L i pk dxp )] = X(R i pkl dx p ) = R i pkl dx p (X) = R i pkl X p, kl Xi lk = R i pkl Xp T p kl Xi i.e., the known identity in local coodinates. So, we have poved the following theoem. Theoem.. In the space L N, with non-symmetic affine connection by equation (.) fo = = the fist Ricci type identity fo a vecto is given. That identity can be witten in foms (.), (.6), (.9), hee R is given by (.) and by (.8). The coesponding identity in local coodinates is (.0)... By using equation (.) and the condition X(ω) = ω(x) F(M N ), we obtain the equation analogous to (.) (X and ω have changed the oles): (.) ( ω ω)(x) = [, ][ω(x)] ω( X X). Definition.. Equations (.) fo, {, } ae Ricci type identities fo a covecto in L N. Equation (.) can be obtained also by consideation of the expession on the left-hand side in (.). The known Ricci identity fo a covaiant vecto in local p, R

5 ON RICCI TPE IDENTITIES coodinates can be obtained fom (.) by substituting ω = ω j x j, X = i, = k, = l : p (.) ω j kl ω j lk = R jkl ω p T p kl ω j p. So, the following theoem is valid. Theoem.. In the space L N, with non-symmetic affine connection, by equation (.) fo = =, the fist Ricci type identity fo a covecto is given. The coesponding identity in local coodinates is (.)... Fo = = fom (.) is obtained (.) ( X X)(ω) = [, ][X(ω)] X( ω ω). Fom hee (.) R(X;, )(ω) = R(ω;, )(X), whee R is expessed by analogously to (.8) and R is given in (.). Supassing to local coodinates, fom (.) one obtains (.) X i kl Xi lk = R i pkl Xp + T p kl Xi and also equations simila to (.), (.) (fo a covecto). Thus, we state Theoem.. In the space L N with non-symmetic affine connection, defined by (.), the second Ricci type identity fo a vecto is given by equation (.). The coesponding identity in local coodinates is (.). p,. Identities fo a vecto and covecto obtained by combinations of both connections.. Putting =, = into (.), we get the identity (.) ( X X)(ω) = [, ][X(ω)] ( ω ω)(x). and fom (.6) (.) ( X X + [,] X)(ω) = ( ω ω + [,] ω)(x) Analogously to (.), let us put (.) R(ω;, ) = ω ω + ω X ω (M N )

6 0 MINČIĆ and (.) becomes (.) ( R(X;, ) + X + X [,] X)(ω) Because of = ( R(ω;, ) + ω + ω [,] ω)(x). ( X X + [,] X)(ω) = ( X +[, ]X)(ω) and = ( X X)(ω) ( ω ω + [,] ω)(x) = ( ω +[, ]ω)(x) = ( ω ω)(x) = [ω(x)] ω( X) [ω(x)] + ω( X) = ( X X)(ω), we see that the ight-hand sides of these equations ae identical and fom (.): (.) R(X;, )(ω) = R(ω;, )(X)... If we put X = X j j, = k, = l, ω = dx i, equation (.) will be witten in local coodinates as follows. Fo the left-hand side L we have L = ( l k X k l X)(dx i ) (.6) whee = [ l (X j k j) k (X j l j)](dx i ) = [(X j k ),l j + X j k Lp lj p (X j l ),k j X j l Lp jk p](dx i ) = (X i k ),l + X j k Li lj (X i l ),k X j = X i k l Xi l k Lp lk (Xi X i k X i l p Xi l = (Xi k ),l + X p k = (Xi l ),k + X p l Li jk p ) = Xi k l Xi l k Li lp Xi p Lp lk, l Li pk Xi p Lp lk. k Lp lk T i spx s,

7 ON RICCI TPE IDENTITIES... Fo the ight-hand side one obtains (.7) R = ( l k dx i k l dx i )(X) = (L i pk,l dxp L i pk Lp ls dxs + L i lp,k dxp + L i lp Lp sk dxs )(X) = (L i pk,l Li sk Ls lp Li lp,k + Li ls Ls pk )Xp. By vitue of (.6) and (.7), fom L = R it is (.8) X i k l Xi l k = R i pkl Xp, and, analogously to that exposed above, fo a covaiant vecto ω it is obtained p (.9) ω j ω j l k = R k l jkl ω p... Intoducing R(X;, ) into (.) by vitue of (.6) and defining R(ω;, ) accoding to (.0) equation (.) gives (.) R(ω;, ) = ω ω + ω X ω (M N ), As in the case of ( R(X;, ) + X + X [,] X)(ω) = ( R(ω;, ) + ω + ω [,] ω)(x). R, we obtain R(X;, )(ω) = R(ω;, )(X). Putting X = j, = k, = l, ω = dx i and taking into consideation (.0), we get R p jkl p(dx i ) = [ l ( L i pkdx p ) k ( L i lpdx p ) + L p kl pdxi L p kl pdxi ]( j ), fom whee fo R the value (.) is obtained. In view of (.0), (.) it is and, using (.8) and (.9), we obtain (.) (.) X i k l Xi l ω j k l ω j R p jkl R p jkl = T i pj T p kl. i k = R pkl Xp + Tps i T kl s Xp, l Now, we can state the following theoem p k = R jkl ω p + T p sj T klω s p.

8 MINČIĆ Theoem.. In the space L N with two non-symmetic affine connections,, linked by equation (.), the thid Ricci type identity fo a vecto is given by equation (.). This identity can be witten also in foms (.), (.) and (.). Fom (.8), fo =, =, one obtains the thid Ricci type identity fo a covecto. The coesponding identities in coodinates ae (.8), (.9), (.) and (.)... In ode to obtain an identity in which R appeas, let us stat fom the expession which appeas in (.7). So, ( X + X X X)(ω) (.) = { [X(ω)] [X( ω)] [X( ω)] + X( ω) + [X(ω)] [X( ω)] [X( ω)] + X( ω) [X(ω)] + [X( ω)] + [X( ω)] X( ω) that is (.) [X(ω)] + [X( ω)] + [X( ω)] X( ω) } (ω), ( X + X X X)(ω) = [, ][X(ω)] + X( ω + ω ω ω). Definition.. Equation (.) we call the combined Ricci type identity fo a vecto in L N. Using (.7), fom (.) it is obtained (.) R(X;, )(ω)+( 0 [, ] X)(ω) = [, ][ω(x)]+ ( ω+ ω ω ω)(x). Fom ( 0 [, ] X)(ω) = [, ][X(ω)] X( 0 [, ] ω) = [, ][ω(x)] + ( 0 [,] ω)(x), we find the fist addend of the ight-hand side and substitute into (.). So, R(X;, )(ω) + ( 0 [, ] X)(ω) = ( 0 [, ] X)(ω) + ( 0 [, ] ω)(x) + ( ω + ω ω ω)(x),

9 ON RICCI TPE IDENTITIES... whee R is given in (.7). Denoting (.6) R(ω;, ) = ( ω + ω ω ω + [,] ω + [,] ω) the pevious equation gives (.7) R(X;, )(ω) = R(ω;, )(X). Substituting hee X = j, = k, = l, ω = dx i and taking into consideation (.6), fo R i jkl (.) is obtained. Remak.. We see that elation between R and R is not of the fom elating to R, R, =,,,. In fact using the coesponding values fom [] R = 0 R(X;, ) + τ(τ(x, ), ) + τ(τ(x, ), ), 8 R = ( X + X X X + [,] X + [,] X) = 0 R(X;, ) τ(τ(x, ), ) τ(τ(x, ), ). We conclude that R 0 R = 8 R and So, we have R(X;, )(ω) = R(ω;, )(X) = R(ω;, )(X). Theoem.. In the space L N with two non-symmetic affine connections,, linked accoding to (.), by equation (.) the combined Ricci type identity fo a vecto is given. Some othe foms of (.) ae (.) (.7). Fom (.) combined Ricci type identity fo a covecto is obtained: (.8) ( ω + ω ω ω)(x) = ( X + X X X)(ω) + [, ][ω(x)]. Fom (.) and (.8) we obtain the coesponding combined Ricci type identities fo a vecto and covecto espectively local coodinates: (Xi kl + Xi kl Xi l k Xi l k ) = R i pklx p, (ω j + ω j ω j l ω j l k) = ( R R) 0 kl kl k p jkl ω p, whee 0 R i jkl is defined by 0 L i jk = (Li jk + Li jk ), i.e., by symmetic connection coefficients. 8

10 MINČIĆ Definition.. The objects on M N, we call dual cuvatue tensos in elation to R. R, ( =,..., ), defined by R : X X X X. Identities fo a tenso field t of the type (, s).. Let us conside a tenso field of the type (, s), which will be denoted ts t, i.e., conside a mapping t s : (X ) (X ) s F(M N ). So, ts (ω,..., ω, X,..., X s ) F(M N ), is a diffeentiable function on M N. As known, a covaiant deivative t s is also of a type (, s). As in (.), one can conside the expession ( ts ts )(ω,..., ω, X,..., X s )... Let us examine neae the case (, s) = (, ), i.e., t t. We have Denoting t = t, we have ( t )(ω, ω ; X) = [t(ω, ω ; X] t( ω, ω ; X) ( t)(ω, ω ; X) = ( t)(ω, ω ; X) t(ω, ω ; X) t(ω, ω ; X). = [t(ω, ω ; X] t( ω, ω ; X) t(ω, ω ; X) t(ω, ω ; X) () = [( t)(ω, ω ; X] ( t)( ω, ω ; X) ( t)(ω, ω ; X) ( t)(ω, ω ; X) = { [t(ω, ω ; X] t( ω, ω ; X) t(ω, ω ; X) t(ω, ω ; X)} () { [t( ω, ω ; X] t( ω, ω ; X) t( ω, ω ; X) t( ω, ω ; X)} { [t(ω, ω ; X] t( ω, ω ; X) t(ω, ω ; X) t(ω, ω ; X)} { [t(ω, ω ; X] t( ω, ω ; X) t(ω, ω ; X) t(ω, ω ; X)}. wheefom (.) ( t t )(ω, ω ; X) = [, ][ t (ω, ω ; X)] = t ( ω ω, ω ; X) t (ω, ω ω ; X) t (ω, ω ; X X).

11 ON RICCI TPE IDENTITIES..... In the geneal case, stating fom (.) we get ( ts )(ω,..., ω ; X,..., X s ) = [ t s (ω,..., ω ; X,..., X s )] ts (ω,..., ω i, ω i, ω i+,..., ω ; X,..., X s ) i= s ts (ω,..., ω ; X,..., X j, X j, X j+,..., X s ), j= ( ts ts )(ω,..., ω ; X,..., X s ) = [, ][ s t (ω,..., ω ; X,..., X s )] (.) ts (ω,..., ω i, ω i ω i, ω i+,..., ω ; X,..., X s ) i= s ts (ω,..., ω ; X,..., X j, X j X j, X j+,..., X s ). j= Conside some paticula cases, obtained fom (.). Fo example: ) Fo = =, =, s = 0 that is fo t 0 = X, fom (.) coesponding identity is obtained, i.e., in coodinates, (.0), and fo =, s = 0 it follows (.), espectively (.) ) Fo = =, =, s = 0 analogous elation (.), and equations coesponding to (.) and (.) ae obtained. ) Fo =, =, =, s = 0 we obtain (.) and fo = 0, s = the coesponding equation follows, whee the oles of X and ω ae exchanged. ) Fo =, s =, elation (.) follows... Identities (.) can be witten so that in them cuvatue tensos figue explicitly. Fo example, fo = = = s =, we have ( t t )(ω; X) = [, ][ t(ω; X)] t ( ω ω; X) t (ω; X X) = [, ][ t(ω; X)] t( R(ω;, ) [,]ω; X) t(ω; R(X;, ) [,]X) = [, ][ t(ω; X)] t( R(ω;, ); X) + t( [,]ω; X) t(ω; R(X;, )) + t(ω; [,]X). In view of (.) it is ( [, ] t )(ω; X) = [, ][ t (ω; X)] + t ( [,]ω; X) + t (ω; [,]X),

12 6 MINČIĆ the pevious equation gives the identity (.) ( t t )(ω; X) = ( [, ] t )(ω; X)] t( R(ω;, ); X) t(ω; R(X;, )). Heefom, in the local coodinates one obtains t i j kl ti j lk = R i pkl tp j R p jkl ti p T p kl ti j p. Finally, fom the exposed, the following theoem is valid. Theoem.. In the space L N with two non-symmetic connections,, linked with equation (.), equation (.) epesents geneal Ricci type identity fo a tenso t s of the type (, s). The equations obtained peviously fo a vecto and a covecto, also (.), ae paticula cases of (.). In (.) we see how the quantities R and Refeences R can be intoduced.. L. S. Das, R. Nivas, S. Ali, M. Ahmad, Study of submanifolds immesed in a manifolds with quate symmetic semi symmetic connection, Tenso N.S. 6 (00), T. Imai, Notes on semi-symmetic metic connections, Tenso N.S. (97), S. M. Minčić, Ricci identities in the space of non-symmetic affine connexion, Mat. Vesnik 0()() (97), 6 7.., On cuvatue tensos and pseudotensos of the space with non-symmetic affine connection, Math. Balkan.?? (97), 7 0 (in Russian).., Cuvatue tensos of the space of non-symmetic affine connexion, obtained fom the cuvatue pseudotensos, Mat. Vesnik (8) (976),. 6., New commutation fomulas in the non-symetic affine connexion space, Publ. Inst. Math. (Beogad) (N.S) (6) (977), , Independent cuvatue tensos and pseudotensos of spaces with non-symmetic affine connexion, Coll. Math. Soc. János Bolyai, Dif. Geom., Budapest (Hungay), (979), , On cuvatue tensos of non-symmetic affine connection, Acta Comment. Univ. Tatu. Math. 9 (00), 0. 9., Some chaacteistics of cuvatue tensos of nonsymmetic affine connexion, Novi Sad J. Math. 9() (999), S. M. Minčić, Lj. S. Velimiović, Diffeential Geomety of Manifolds, Faculty of Science and Mathematics, Univ. of Niš, 0 (in Sebian).. C. Nitescu, Bianchi identities in a non symmetic connection space, Bul. Inst. Polit. Iasi Sec., Mat. Mec. teo. Fiz. 0()(-) (97),. H. D. Pande, Decomposition of cuvatue tenso in R ecuent Finsle space with nonsymmetic connection (I), Pog. Math. (,) (988), M. Pvanović, Une connexion non-symmetique associé á l éspace Riemannien, Publ. Inst. Math. 0() (970), 6.., Fou cuvatue tensos of non-symmetic affine connexion, Poc. Conf. 0 yeas of Lobachevski geomety", Kazan 976, Moscow 997, 99 0 (in Russian).. U. P. Singh, On elative cuvatue tensos in the subspace of a Riemannian space, Revne Fac. Sci. Instambul (968), 69 7.

13 ON RICCI TPE IDENTITIES K. ano, On semi-symmetic metic connection, Revne Roum. Math. Pues Appl. (970), N. oussef, A. Sid-Ahmed, Linea connection and cuvatue tensos of the geomety of paallelisable manifolds, Repots Math. Phys. 60() (007), S. M. Minčić, On cuvatue tensos obtained by two non-symmetic affine connections, submitted. Faculty of Science and Mathematics Univesity of Niš Niš Sebia svetislavmincic@yahoo.com