A Class of Riemannian Manifolds with Special Form of Curvature Tensor

Abstract

Abstract Riemannian curvature is determined by the Ricci curvature by a special formula to introduce a special class of Riemannian manifolds which called Jawarneh manifold. Some geometric properties of Jawarneh manifold have been derived and a non-trivial example is obtained to prove the existence. Keywords: Riemannian manifold; flat manifold; Einstein manifold; symmetric manifold; Ricci symmetric manifold; conformally flat manifold; cosmology. Mathematics Subject Classification (2010): 53C25 (primary); 53C21 (secondary). 1 Introduction It is well known that on a Riemannian manifold (M n , g) the curvature tensor vanishes for n = 1, and it is proportional to the metric g for n = 2. But for n = 3 the curvature tensor of a Riemannian manifold is proportional to the Ricci tensor. Hence without loss of generality we can define a special type of Riemannian manifolds by considering a Riemannian manifold (M n , g) (n>2) with non-vanishing Ricci tensor S and its curvature tensor R satisfies the relation; R(X,Y,Z) = k[S(Y,Z)X−S(X,Z)Y], (1.1) Original Research Article