On a Generalization of the Spaces of Quasi-Constant Curvature

S.S.Chern [3] has studied some properties of the curvature and characteristic classes of a Riemannian manifold (Mg) whose Riemann curvature tensor R satisfies the relation (R(Z,W)Y,X) S(X,Z)S(Y,W) S(X,W)S(Y,Z), (I) where S is a symmetric tensor of type (0,2), and > the inner product with respect to g. It is known that the spaces of quasi—constant curvature [I] are of form (I) In physical literature, the 4—dimensional Lorentzian manifolds of quasi—constant curvature are called infinitesimally spatially isotropic, they are conformally flat solutions of Einstein's gravitational field equations for the perfect fluid matter [5]. In this note we shall consider a class of Riemannian manifolds which is a natural generalization of the spaces of quasi—constant curvature. These manifolds do not satisfy the relation of form (I). but still have some remarkable properties in their curvature and characteristic classes. Their physical background is the cosmological models with heat now [6]. with an assumption that they are conformally flat.