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

Hari Baskar, R.

Maralabhavi, Y .B.

Shivaprasanna, G. S.

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

Maralabhavi, Y.B.

Shivaprasanna, G.S.

ePrints@Bangalore University

Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 47, 2327 - 2333On Weakly Symmetric and Weakly ConformallySymmetric Spaces admitting Veblen identitiesY. B. MaralabhaviDepartment of MathematicsBangalore UniversityBangalore, Indiaybmbub@yahoo.co.inHari Baskar R.Department of MathematicsChrist UniversityBangalore, IndiaShivaprasanna G. S.Department of MathematicsAmruta Institute of Engineering and Management SciencesBangalore, IndiaAbstractIn the present paper some properties involving curvature tensor,conformal curvature tensor, Ricci tensor and scalar curvature, on weaklysymmetric, weakly conformally symmetric and pseudo symmetric spacesare obtained.Mathematics Subject Classification: 53A40Keywords: Weakly Symmetric, Weakly Conformally symmetric1 IntroductionL.Tamassy and T.Q.Binh [3] have introduced the notion of weakly symmetricand weakly projective symmetric spaces. Based on this work, U.C.De andS.Bandyopadhyaa [7] introduced the notion of weakly conformally symmetricspaces and investigated some properties of such spaces. We consider these2328 Y. B. Maralabhavi, Hari Baskar R. and Shivaprasanna G. S.spaces admitting Veblen identities and conformal Veblen identities [4] anddetermine some other properties.Let Mn be a Riemannian n-dimensional space covered by system of co-ordinate neighbourhoods (U, xi). Suppose gij, Rhijk and Rij denote the localcomponents of the metric tensor, the curvature tensor and the Ricci tensor,respectively, and let R denote the scalar curvature. The non-ﬂat Riemannianspace Mn (n > 2) is called weakly symmetric space if the curvature tensorRhijk satisﬁes the condition [3]Rhijk,l = a1Rhijk + bhRlijk + diRhljk + ejRhilk + fkRhijl ,(1.1)where a, b, d, e, f are 1-forms (non-zero simultaneously) and the ‘,’ denotesthe covariant diﬀerentiation with respect to the metric tensor of the space.An n-dimensional weakly symmetric space Mn is denoted by (WS)n. Suchspaces are studied by M.Pranovic [5], T.Q.Binh [6] and others. U.C.De andS.Bandyopadhyay [8] proved that the associated 1-forms d and f in (1.1) areidentical with b and e, respectively. Hence the condition (1.1) of (WS)n be-comesRhijk,l = alRhijk + bhRlijk + biRhljk + ejRhilk + ekRhijl .(1.2)Further, the space Mn is called pseudo symmetric ifRhijk,l = 2alRhijk + ahRlijk + aiRhljk + ajRhilk + akRhijl .An n-dimensional non conformally ﬂat Riemannian space Mn (n > 3) is calledweakly conformally symmetric if its conformal curvature tensor Chijk, given byChijk = Rhijk +1n− 2(ghkRij − ghjRik + gijRhk − gikRhj)(1.3)+R(n− 1)(n− 2)(ghkgij − ghjgik),satisﬁes the conditionChijk,l = alChijk + bhClijk + diChljk + ejChilk + fkChijl(1.4)where a, b, d, e, f are associated 1-forms (non zero simultaneously). A weaklyconformally symmetric space Mn is denoted by (WCS)n. As in case of (WS)nit is proved that d and f are identical with b and e respectively. So, (1.4)reduces toChijk,l = alChijk + bhClijk + biChljk + ejChilk + ekChijl(1.5)Weakly symmetric and weakly conformally symmetric spaces 2329for weakly conformally symmetric spaces. The space Mn is called pseudoconformally symmetric ifChijk,l = 2alChijk + ahClijk + aiChljk + ajChilk + akChijl .The conformal curvature tensor satisﬁes the conditions:Chijk + Chjki + Chkij = 0 ,(1.6)Crrjk = Crirk = Crijr = 0 ,(1.7)andChijk = −Chikj = Cihkj = Ckjih .(1.8)A space is said to be quasi conformally ﬂat ifC ′hijk = 0,(1.9)whereC ′hijk = aZhijk + b (ghkGij − gjkGik + gijGhk − gikGhj) ,(1.10)with a, b as arbitrary constants andZhijk = Khijk − Rn (n− 1) (ghkgij − gkighj) , Gij = Rij −Rngij.The Veblen identities and conformal Veblen identities in Mn are given by [4]:V hijkl = Rhijk,l + Rhkil,j + Rhlkj,i + Rhjli,k = 0(1.11)andW hijkl = Chijk,l + Chkil,j + Chlkj,i + Chjli,k(1.12)− 1(n− 3)ghm{(gjmCpkil,p + gkmCpjli,p + gimCplkj,p + glmCpijk,p)−(gikCpmlj,p + gijCpmkl,p + gklCpmji,p + gljCpmil,p)} = 0 .If the Ricci tensor satisﬁes the conditionRij =Rngij ,(1.13)then Mn is called Einstein space.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,Rhijk,l + Rhikl,j + Rhilj,k = 0 ,(2.1)we getβlRhijk + βjRhikl + βkRhilj = 0 ,(2.2)where we have putβl = al − 2el(2.3)and used the relationsRhijk + Rhjki + Rhkij = 0(2.4)andRhijk = −Rihjk = Rihkj(2.5)satisﬁed by the curvature tensor ﬁeld. By transvecting (2.2) by glh and using(2.5), we getβhRhijk = βkRji − βjRki.(2.6)Now transvection of (2.6) with gij givesβhRhk =βkR2.(2.7)If Mn is an Einstein space, then (2.7) reduces to(n− 2)βkR = 0.(2.8)Now write the Veblen identity (1.11) in the formRhijk,l + Rhkil,j + Rhlkj,i + Rhjli,k = 0(2.9)and use (1.2) to getαiRhlkj + αjRhkil + αkRhjli + αlRhijk = 0,(2.10)where we have putαi = ai − (bi + ei) ,(2.11)Weakly symmetric and weakly conformally symmetric spaces 2331and used the results (2.4) and (2.5) . Transvecting (2.10) by gkhand using(2.5), we getαhRhjli = αiRlj − αlRij.(2.12)Now by transvecting (2.12) with gil and using that Mn is an Einstein space,weget(n− 2)αiR = 0.(2.13)In view of (2.8) and (2.13) we haveTheorem 2.1 The scalar curvature of weakly symmetric Einstein Rieman-nian space Mn is zero provided the 1-form a in (1.2) is neither 2e nor b + eRemark 2.2 As per G.Herglotz [1] the scalar curvature of Einstein spaceMn is constant. But that constant is necessarily zero if Mn is weakly symmetricin which a = 2e, b + e for the 1-forms a, b , e used in (1.2).Suppose R = 0 in Mn, then from (2.8) and (2.13) we see that βi = 0 andαi = 0, which in turn indicate a = 2e,a = b + e and hence b = e. Hence Mnreduces to pseudosymmetric. So, we state the following:Theorem 2.3 A weakly symmetric Einstein space with non zero scalar cur-vature is pseudo symmetric.3 Weakly conformally symmetric space (WCS)n.By using (1.5) in (1.12), we getalChijk + bhClijk + biChljk + ejChilk + ekChijl+ajChkil + bhCjkil + bkChjil + eiChkjl + elChkij+aiChlkj + bhCilkj + blChikj + ekChlij + ejChlki+akChjli + bhCkjli + bjChkli + elChjki + eiChjlk−(ap + bpn− 3)[{glhCpijk + gjhCpkil + gihCplkj + gkhCpjli}2332 Y. B. Maralabhavi, Hari Baskar R. and Shivaprasanna G. S.− {gklCphji + gljCphik + gjiCphkl + gikCphlj}] = 0 .In view of (1.6), (1.7) and (1.8) above result reduces toαlChijk + αjChkil + αiChlkj + αkChjli −(ap + bpn− 3)[{glhCpijk + gjhCpkil(3.1)+gihCplkj + gkhCpjli} − {gklCphji + gljCphik + gijCphkl + gikCphlj}] = 0 ,where αl = al − (bl + el). Contracting (3.1) by ghi and using (1.6), (1.7) and(1.8), we getλpCplkj = 0, where λp = 2bp + ep .(3.2)Transvecting (3.1) by λl and using (1.7), (1.8) and (3.2), we get(λlαl)Chijk =(al + bln− 3)[λhClijk + λkClhij + λjClhki].(3.3)Transvecting (3.3) with λh and using (3.2), we get(λhλh)(al + bln− 3)C lijk = 0and hence(al + bl)Clijk = 0 .(3.4)Now the equation (3.3), in view of (3.4), reduces to(λlαl)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 λlαl = 0, whereλl = 2bl + el and αl = al − (bl + el).Suppose λlαl = 0. Then Chijk = 0. Now, by usingProposition A:[2] A quasi-conformally ﬂat space is either conformally ﬂator Einstein.We state the following:Theorem 3.2 A quasi conformally flat weakly conformally symmetric space(WCS)n, (n > 3) with λlαl = 0 is Einstein space.Weakly symmetric and weakly conformally symmetric spaces 2333References[1] G.Herglotz, Zur Einsteinschen Gravitationstheorie, Sitzungsber,Sachs.Gesellsch. Wiss. Leipzig, 68 (1961), 199 - 203.[2] Krishna Amur and Y.B. Maralabhavi, On Quasi-Conformally Flat Spaces,the TENSOR (New Series) 31(2) (1977), 194-198.[3] L.Tamassy and T.Q.Binh, On weakly symmetric and weakly projectivesymmetric Riemannian manifolds,Coll.Math,Soc.J.Bolyai, 56 (1992), 663- 670.[4] M.D.Upadhyay, Conformal curvature identities,Tensor N.S, 21 (1970), 33- 36.[5] M.Pravanovic, On weakly symmetric Riemannian manifolds,Publ. Math.Debrecen, 46 (1995), 19 - 25.[6] T.Q.Binh, On weakly symmetric Riemannian spaces,Publ.Math.Debrecen,42 (1993), 103 - 107.[7] U.C.De and S.Bandyopadhyay, On weakly conformally symmetricspaces,Publ.Math.Debrecen, 57 (2000), 71 - 78.[8] U.C.De and S.Bandyopadhyay, On weakly symmetric Riemannianspaces,Publ.Math.Debrecen, 54 (1999), 377 - 381.Received: June, 2011

On weakly symmetric and weakly conformally symmetric spaces admitting Veblen identities.

On Weakly Symmetric and Weakly Conformally Symmetric Spaces Admitting Veblen Identities

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On Weakly Symmetric and Weakly Conformally Symmetric Spaces Admitting Veblen Identities

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