On geodesic mappings of Riemannian spaces with cyclic Ricci tensor

An n-dimensional Riemannian space V-n is called a Riemannian space with cyclic Ricci tensor [2, 3], if the Ricci tensor R-ij satisfies the following condition R-ij,R- k vertical bar R-jk,R- i vertical bar R-ki,R- j = 0, where R-ij the Ricci tensor of V-n, and the symbol "," denotes the covariant derivation with respect to Levi-Civita connection of V-n. In this paper we would like to treat some results in the subject of geodesic mappings of Riemannian space with cyclic Ricci tensor. Let V-n = (M-n, g(ij)) and (V) over bar (n) = (M-n, (g) over bar (ij)) be two Riemannian spaces on the underlying manifold M-n. A mapping V-n -> (V) over bar (n) is called geodesic, if it maps an arbitrary geodesic curve of V-n to a geodesic curve of (V) over bar (n).[4] At first we investigate the geodesic mappings of a Riemannian space with cyclic Ricci tensor into another Riemannian space with cyclic Ricci tensor. Finally we show that, the Riemannian-Einstein space with cyclic Ricci tensor admit only trivial geodesic mapping.