On Weakly Symmetric and Weakly Conformally Symmetric Spaces admitting Veblen identities

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 47, On Weakly Symmetric and Weakly Conformally Symmetric Spaces admitting Veblen identities Y. B. Maralabhavi Department of Mathematics Bangalore University Bangalore, India Hari Baskar R. Department of Mathematics Christ University Bangalore, India Shivaprasanna G. S. Department of Mathematics Amruta Institute of Engineering and Management Sciences Bangalore, India Abstract In the present paper some properties involving curvature tensor, conformal curvature tensor, Ricci tensor and scalar curvature, on weakly symmetric, weakly conformally symmetric and pseudo symmetric spaces are obtained. Mathematics Subject Classification: 53A40 Keywords: Weakly Symmetric, Weakly Conformally symmetric 1 Introduction L.Tamassy and T.Q.Binh [3] have introduced the notion of weakly symmetric and weakly projective symmetric spaces. Based on this work, U.C.De and S.Bandyopadhyaa [7] introduced the notion of weakly conformally symmetric spaces and investigated some properties of such spaces. We consider these

2 2328 Y. B. Maralabhavi, Hari Baskar R. and Shivaprasanna G. S. spaces admitting Veblen identities and conformal Veblen identities [4] and determine some other properties. Let M n be a Riemannian n-dimensional space covered by system of coordinate neighbourhoods (U, x i ). Suppose g ij, R hijk and R ij denote the local components of the metric tensor, the curvature tensor and the Ricci tensor, respectively, and let R denote the scalar curvature. The non-flat Riemannian space M n (n>2) is called weakly symmetric space if the curvature tensor R hijk satisfies the condition [3] (1.1) R hijk,l = a 1 R hijk + b h R lijk + d i R hljk + e j R hilk + f k R hijl, where a, b, d, e, f are 1-forms (non-zero simultaneously) and the, denotes the covariant differentiation with respect to the metric tensor of the space. An n-dimensional weakly symmetric space M n is denoted by (WS) n. Such spaces are studied by M.Pranovic [5], T.Q.Binh [6] and others. U.C.De and S.Bandyopadhyay [8] proved that the associated 1-forms d and f in (1.1) are identical with b and e, respectively. Hence the condition (1.1) of (WS) n becomes (1.2) R hijk,l = a l R hijk + b h R lijk + b i R hljk + e j R hilk + e k R hijl. Further, the space M n is called pseudo symmetric if R hijk,l =2a l R hijk + a h R lijk + a i R hljk + a j R hilk + a k R hijl. An n-dimensional non conformally flat Riemannian space M n (n>3) is called weakly conformally symmetric if its conformal curvature tensor C hijk, given by (1.3) C hijk = R hijk + 1 n 2 (g hkr ij g hj R ik + g ij R hk g ik R hj ) R + (n 1)(n 2) (g hkg ij g hj g ik ), satisfies the condition (1.4) C hijk,l = a l C hijk + b h C lijk + d i C hljk + e j C hilk + f k C hijl where a, b, d, e, f are associated 1-forms (non zero simultaneously). A weakly conformally symmetric space M n is denoted by (WCS) n. As in case of (WS) n it is proved that d and f are identical with b and e respectively. So, (1.4) reduces to (1.5) C hijk,l = a l C hijk + b h C lijk + b i C hljk + e j C hilk + e k C hijl

3 Weakly symmetric and weakly conformally symmetric spaces 2329 for weakly conformally symmetric spaces. conformally symmetric if The space M n is called pseudo C hijk,l =2a l C hijk + a h C lijk + a i C hljk + a j C hilk + a k C hijl. The conformal curvature tensor satisfies the conditions: (1.6) C h ijk + C h jki + C h kij =0, (1.7) C r rjk = C r irk = C r ijr =0, and (1.8) C hijk = C hikj = C ihkj = C kjih. A space is said to be quasi conformally flat if (1.9) C hijk =0, where (1.10) C hijk = az hijk + b (g hk G ij g jk G ik + g ij G hk g ik G hj ), with a, b as arbitrary constants and Z hijk = K hijk R n (n 1) (g hkg ij g ki g hj ), G ij = R ij R n g ij. The Veblen identities and conformal Veblen identities in M n are given by [4]: (1.11) and (1.12) V h ijkl = R h ijk,l + R h kil,j + R h lkj,i + R h jli,k =0 W h ijkl = Ch ijk,l + Ch kil,j + Ch lkj,i + Ch jli,k 1 (n 3) ghm {(g jm C p kil,p + g kmc p jli,p + g imc p lkj,p + g lmc p ijk,p ) If the Ricci tensor satisfies the condition (g ik C p mlj,p + g ijc p mkl,p + g klc p mji,p + g ljc p mil,p )} =0. (1.13) R ij = R n g ij, then M n is called Einstein space.

4 2330 Y. B. Maralabhavi, Hari Baskar R. and Shivaprasanna G. S. 2 Weakly symmetric space (WS) n. By using (1.2) in the Bianchi identity, (2.1) R hijk,l + R hikl,j + R hilj,k =0, we get (2.2) β l R hijk + β j R hikl + β k R hilj =0, where we have put (2.3) β l = a l 2e l and used the relations (2.4) R hijk + R hjki + R hkij =0 and (2.5) R hijk = R ihjk = R ihkj satisfied by the curvature tensor field. By transvecting (2.2) by g lh and using (2.5), we get (2.6) β h R h ijk = β k R ji β j R ki. Now transvection of (2.6) with g ij gives (2.7) β h Rk h = β kr 2. If M n is an Einstein space, then (2.7) reduces to (2.8) (n 2) β k R =0. Now write the Veblen identity (1.11) in the form (2.9) R hijk,l + R hkil,j + R hlkj,i + R hjli,k =0 and use (1.2) to get (2.10) α i R hlkj + α j R hkil + α k R hjli + α l R hijk =0, where we have put (2.11) α i = a i (b i + e i ),

5 Weakly symmetric and weakly conformally symmetric spaces 2331 and used the results (2.4) and (2.5). (2.5), we get Transvecting (2.10) by g kh and using (2.12) α h R h jli = α i R lj α l R ij. Now by transvecting (2.12) with g il and using that M n is an Einstein space,we get (2.13) (n 2) α i R =0. In view of (2.8) and (2.13) we have Theorem 2.1 The scalar curvature of weakly symmetric Einstein Riemannian space M n is zero provided the 1-form a in (1.2) is neither 2e nor b + e Remark 2.2 As per G.Herglotz [1] the scalar curvature of Einstein space M n is constant. But that constant is necessarily zero if M n is weakly symmetric in which a 2e, b + e for the 1-forms a, b, e used in (1.2). Suppose R 0inM n, then from (2.8) and (2.13) we see that β i = 0 and α i = 0, which in turn indicate a =2e,a = b + e and hence b = e. Hence M n reduces to pseudosymmetric. So, we state the following: Theorem 2.3 A weakly symmetric Einstein space with non zero scalar curvature is pseudo symmetric. 3 Weakly conformally symmetric space (WCS) n. By using (1.5) in (1.12), we get a l C hijk + b h C lijk + b i C hljk + e j C hilk + e k C hijl +a j C hkil + b h C jkil + b k C hjil + e i C hkjl + e l C hkij +a i C hlkj + b h C ilkj + b l C hikj + e k C hlij + e j C hlki +a k C hjli + b h C kjli + b j C hkli + e l C hjki + e i C hjlk ap + b p [{g lh C p ijk n 3 + g jhc p kil + g ihc p lkj + g khc p jli }

6 2332 Y. B. Maralabhavi, Hari Baskar R. and Shivaprasanna G. S. {g kl C p hji + g ljc p hik + g jic p hkl + g ikc p hlj }]=0. In view of (1.6), (1.7) and (1.8) above result reduces to (3.1) α l C hijk + α j C hkil + α i C hlkj + α k C hjli ap + b p [{g lh C p ijk n 3 + g jhc p kil +g ih C p lkj + g khc p jli } {g klc p hji + g ljc p hik + g ijc p hkl + g ikc p hlj }]=0, where α l = a l (b l + e l ). Contracting (3.1) by g hi and using (1.6), (1.7) and (1.8), we get (3.2) λ p C p lkj = 0, where λ p =2b p + e p. Transvecting (3.1) by λ l and using (1.7), (1.8) and (3.2), we get λ l al + b l [λh (3.3) α l Chijk = Cijk l n 3 + λ kchij l + λ jchki] l. Transvecting (3.3) with λ h and using (3.2), we get λ h a l + b l λ h Cijk l n 3 =0 and hence (3.4) (a l + b l ) C l ijk =0. Now the equation (3.3), in view of (3.4), reduces to ( λ l α l ) Chijk =0. So we state the following: Theorem 3.1 A weakly conformally symmetric space (WCS) n, (n >3), is conformally flat if the 1-forms a, b, c satisfy the condition λ l α l 0, where λ l =2b l + e l and α l = a l (b l + e l ). Suppose λ l α l = 0. Then C hijk 0. Now, by using Proposition A:[2] A quasi-conformally flat space is either conformally flat or Einstein. We state the following: Theorem 3.2 A quasi conformally flat weakly conformally symmetric space (WCS) n, (n >3) with λ l α l =0is Einstein space.

7 Weakly symmetric and weakly conformally symmetric spaces 2333 References [1] G.Herglotz, Zur Einsteinschen Gravitationstheorie, Sitzungsber, Sachs.Gesellsch. Wiss. Leipzig, 68 (1961), [2] Krishna Amur and Y.B. Maralabhavi, On Quasi-Conformally Flat Spaces, the TENSOR (New Series) 31(2) (1977), [3] L.Tamassy and T.Q.Binh, On weakly symmetric and weakly projective symmetric Riemannian manifolds,coll.math,soc.j.bolyai, 56 (1992), [4] M.D.Upadhyay, Conformal curvature identities,tensor N.S, 21 (1970), [5] M.Pravanovic, On weakly symmetric Riemannian manifolds,publ. Math. Debrecen, 46 (1995), [6] T.Q.Binh, On weakly symmetric Riemannian spaces,publ.math.debrecen, 42 (1993), [7] U.C.De and S.Bandyopadhyay, On weakly conformally symmetric spaces,publ.math.debrecen, 57 (2000), [8] U.C.De and S.Bandyopadhyay, On weakly symmetric Riemannian spaces,publ.math.debrecen, 54 (1999), Received: June, 2011