Decomposition of Riemannian Curvature Tensor Field and Its Properties
James Gathungu Gicheru *
Department of Mathematics, Ngiriambu Girls High School, P.O.Box 46-10100, Kianyaga, Kenya.
Cyrus Gitonga Ngari
Department of Mathematics, Computing and Information Technology, School of Pure and Applied Sciences, University of Embu, P.O.Box 6-60100, Embu, Kenya.
*Author to whom correspondence should be addressed.
Abstract
Decomposition of recurrent curvature tensor fields of R-th order in Finsler manifolds has been studied by B. B. Sinha and G. Singh [1] in the publications del’ institute mathematique, nouvelleserie, tome 33 (47), 1983 pg 217-220. Also Surendra Pratap Singh [2] in Kyungpook Math. J. volume 15, number 2 December, 1975 studied decomposition of recurrent curvature tensor fields in generalised Finsler spaces. Sinha and Singh [3] studied decomposition of recurrent curvature tensor fields in a Finsler space.
In this paper we study the Riemannian Curvature tensor with its properties its decomposition of the Riemannian curvature tensor and its properties. This raises important question: in Riemannian manifold , is it possible to decompose Riemannian curvature tensor of rank four, get another tensor of rank two and study its properties?
Keywords: Riemannian curvature tensor, decomposition