INTERNAL REPORT 1. INTRODUCTION (Limited distribution) Let 7 be a linear connection in a Riemannian manifold M of n (>2) such that it is expressed locally as follows [6];
International Atomic Energy Agency and (1.1) United Nations Educational Scientific and Cultural Organization where <. f are Christoffel symbols, U,, and V,. are,respectively, symmetric INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS I"1/ J1 " _h th h and skew-symmetric parts of an arbitrary tensor T , U7 • U. g , V, • V g hi h Ji J jt J jt and P 1B a 1-form such that g Jp, ° p . J V r,, given by (l.l) is a quarter-symmetric connection with torsion SOME CURVATURE PROPERTIES OF QUARTER SYMMETRIC METRIC COMMECTIOHS • tensor I P( - ^p, and curvature tensor R ^ expressible as [6]
S.C. Rastogi •• International Centre for Theoretical Physics, Trieste, Italy.
(1.2)
t th ABSTRACT where p^ •> VjPl - UJtp pi + | UjjP* pt> pj - Pug and -j|'k means interchange of indices J and k and subtraction. A linear connection r , with torsion tensor T.p. - T.p , where T J A J 1 1 J J is an arbitrary (1,1) tensor field and p, is a 1-form,has been called a 2. SYMMETRIC T quarter-symmetric connection by Golyb [3]. Some properties of such connections If T,. is symmetric V,. • 0 and (1.1) and (1.2), respectively,reduce to have been studied by Rastogi (53. MiBhra and Pandey [U] and Yano and Imai [6], Ji Ji In this paper based on the curvature tensor of quarter-symnetric metric connection we define a tensor analogous to conformed, curvature tensor [2] and (2.1) study some properties of such a tensor. and MIRAMARE - TRIESTE 2.2) August 1°86
From (2.2) it is eaBy to get
• To be submitted for publication. •• Permenent address: Department of Mathematics, University of Nigeria, Nsukka, Nigeria. (2.3)
-2- and Theorem 2.1 A neceaBary and sufficient condition for a Rlenannlan manifold H (£.1*) admitting a quarter-ayinnetric metric connection (2.1) to be con formally flat la t t given by (2.8). where T • T. , p » p. • Prom (2.6) we can see that if •Cfc,lh - C h> M^^ - 0 and How on the basis of R_H »a R we define the following tensor; conversely if " Ot we get C . Hence we have
Theorem 2.2 A necessary and sufficient condition for *C. ... to be equal to conformal curvature tensor la that Wyjij, vanishes identically. If 11 - 0, by multiplying (2.7) by g^ ^ we obtain on simplification which vanishes if \.in " °>
#Ckjih (a.9) If in (2.5) we substitute from (2.2), (2.3) and (2.10 ve obtain Mow ve shall consider some special caaeB (2.6)
- If v. assume T^ - C^ - 2(n^^{tl.2) q. (2.9) where C. ,. is conformal curvatura tenaorfEisenhart (2])aad gives
(2.10)
From (2.10) we obtain the following Corollary to Theorem 2.2.
Corollary 1 If a Rieaannian manifold M admits a quarter-aynmetric metric connection (2.1) for T-. - C.. and satisfies #C. ,,„ • C.,.. . it BIBO satiBfias (2.10). kh Kh kjih Hjin If in (2.10), K? • Corollary 2 If 0, E(i.(£.61 implies If a Rlemanniaa manifold M admits a a,uarter-syi»Betric metric connection (2.1) for T - C , K^ » aS* and also aatisfiee (2.8) kh kh then P vanishes identically. Conversely, if (2.8) la satisfied! (2.6) glveB C.,.. " °> Hence we have -3- Case II - If we assume - ~ K &^, similar to (2.10) Cfch 3. SKEW-SYMMETRIC T ue get If T^ is skew-synmetric u » 0 ^ {la)> u>2) reBpeetively reduce to [6] which laplies a result similar to Corollary 1. (3.1) Caae III - If T^ - K^, (2.9) and (3.2) Now leaving the trivial case of p =0, from (2.12) we can obtain From (3.2) it is easy to obtain •k Corollary 3 (3.3) If a Riemannlan manifold M admits a quarter-symmetric metric connection and (2.1) for T~kfch "Tin ani lf *•* aatisfioB #CkJih = Ck.Hh alao> a nece3aarv »«d sufficient condition for it to be of constant scalar curvature K is that it k (3.U satisfies tCpPt" - 0. If ve substitute from (3.2), (3.3) and (3.1*) in (2.5) we get Caae IV - If T. • og..> for some scalar , and if we also have # ( p. VjO • 0, from equation (2.6) we get C .. « 'iIiih* Conversely, if (3.5) where (2.13) which implies p^9 a - 0, Ph'fca •= 0. Hence, we have Corollary h If a Riemannian manifold M admits a quarter-symmetric metric connection (2.1) for T necesoary and sufficient condition for "C.... to be fch identically equal to conformal curvature tensor C. ... is given by p V a = 0. Case V - If Tkh = gkh. Corollary k easily gives (3.6) obtained for semi-svanetric case by Amur and Pujar [l]. If C e uation kjih * ° 1 (3.5) gives *CkJlh - Nh> conversely if *CkJih " "kjih* Ckjih ° 0> Hence -5- -6- Theorem 3.1 A necessary and sufficient condition for a Rlemannian manifold M admitting a quarter-symmetric metric connection (3.1) to be conforaally flat ia given by and 0, farther if CkUh, - 0 and conversely if "Ckjih ' Theorem 3-2 For this case from (3.6) we shall have A necessary and sufficient condition for »C , to be equal to conforms! curvature tensor is that H. ... vanishes identically. kjln Ji kh If we multiply (3.6) by g g and put H ih - 0, we get '*,) •= O, (3.7) If in a special case p^ is a gradient vector R).1 • K^^ , and R - K, therefore, H,, • 0 and •C , • Cy^* Hence we have Hence we have ih )l in Theorem U.I Corollary 5 c If a Kaehlerian manifold H admits a quarter-symmetric metric connection If a Riemannlan manifold M admits (3.1) and satisfies * 1 1). KAEHLERIAH HAHIFOLD Case II - If U,. - g.. wi V^» F ?, Eq.d.l) gives We now assume that K li I Kaehlerian manifold of 2n (n >, 2) dimension with metric tensor g , and almost complex structure tensor FV satisfying F F s F 8 F F F F g sovwe 5 t-- J* K ts-«ji' ji-- ij- ji" 5 u- Yj-°- while (1.2) gives shall consider some special cases. Case I - If T^ • F^, Eqs. (3.1) and (3.2) respectively reduce to C.7) and From (U.T) we can easily obtain From (k.2) it is easy to get -7- -a- and 2>^ fiEFERENCES (U.9) [l] K. Amur and 5.S. Pujar, "On suUnanifolds of a Riem&nnian manifold In analogy to (2.6) we can obtain admitting a metric semi-Bynmetric connection", Tensor U.S. 3£ (1978) 35-38. [2] L.P. Eisenhart, Rlemannlan Ceometry (Princeton University Press, i960), (U.10) [3] S. Golab, "on senl-synoetric and quarter-eytnetrlc linear connections", Tensor H.5., 29 (1975) 21*9-251*. where [h] U.S. Mishra and S.H. Pandey, "On quarter-aymoetric metric F-conxutioos", Tensor N.S. 3jt. (i960) 1-7. [5] S.C. Rastogi, "On quartar-syanetric metric connection", Comptes RenduB de l'Acad. Bulgar. des Sciences 31. (1978) 8ll-8ll*. [6] K. Yano and T. Imai, "Quarter-sysDetric metric connections and their ^ curvature tensors". Tensor N.S. 38 (1982) 13-18. (U.ll) and From (U.10) we can obtain Theorem U.5 If a Kaehlerian manifold M admits a. quarter-symmetric metric connection (fc.6), a necessary and sufficient condition for «C to be equal to conformal curvature tensor CkJih is that LR1Jh vanishes identically ACKHOWLEDGMEHTS The author would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. -10- -9-