On Generalized Space Matter Tensor

Extending the concept of Petrov tensor, in this article we attempt to introduce generalised space matter tensor [1],[2], [3], [4]. In the Riemannian manifold, it is found that the second Bianchi identity for the generalized space-matter tensor is satisfied if the energy-momentum tensor is of Codazzi type [5]. We study the nature of Riemannian manifolds by imposing curvature restrictions like symmetry, recurrent, weakly symmetry [6], [7], [8] etc. on this generalized Petrov space-matter tensor. We obtain the eigen values of the Ricci tensor S corresponding to the vector fields associated with the various 1− forms.