Conformal mappings preserving the Einstein tensor of Weyl spaces

Abstract

This work contains four chapters. In Chapter 1, the fundamental definitions and properties concerning the Weyl manifolds are given. In this chapter, moreover, the definitions and the basic properties of the mixed curvature tensor and the definitions of the Ricci tensor, the scalar curvature tensor, the Einstein tensor, the conformal curvature tensor, the concircular curvature tensor and the projective curvature tensor (Weyl tensor) of the Weyl manifold are also given. In chapter 2, firstly, the conformal mappings of Weyl Spaces are defined. Then, the relations between connection coefficients, the mixed curvature tensors, the Ricci tensors, the scalar curvature tensors, the Einstein tensors, the concircular curvature tensors and the projective curvature tensors of the Weyl manifolds under the conformal mapping are studied. After that it is proved that the covector field P is a gradient, under the conformal mapping preserving the concircular curvature tensor or the projective curvature tensor of the Weyl manifold. And it is also proved that, if the conformal flat Weyl manifold is a concircular flat Weyl manifold, then the Weyl manifold is an Einstein-Weyl manifold by considering the relation between the concircular curvature tensor and the conformal curvature tensor. After giving the definition and the some properties of the conformal curvature tensor, we close this chapter by obtaining the new invariants under the conformal mapping. In Chapter 3, the necessary and sufficient condition for a conformal mapping between two Weyl manifolds to preserve the Einstein tensor is obtained. Then, it is proved that, if a conformal transformation from a Weyl manifold onto a flat Weyl manifold preserves the Einstein tensor, then both Weyl manifolds are Einstein-Weyl manifolds.