In this article, after giving a necessary and sufficient condition for two Einstein- Weyl manifolds to be in conformal correspondence, we prove that any conformal mapping between such manifolds is generalized concircu...In this article, after giving a necessary and sufficient condition for two Einstein- Weyl manifolds to be in conformal correspondence, we prove that any conformal mapping between such manifolds is generalized concircular if and only if the covector field of the conformal mapping is locally a gradient. Using this fact we deduce that any conformal mapping between two isotropic Weyl manifolds is a generalized concircular mapping. Moreover, it is shown that a generalized concircularly flat Weyl manifold is generalized concircular to an Einstein manifold and that its scalar curvature is prolonged covariant constant.展开更多
It is well known that the Einstein tensor G for a Riemannian manifold defined by Gα^β = 1/2α^β ,Rα^β=g^α^β γ where Rγα and R are respectively the Ricci tensor and the scalar curvature of the manifold, p...It is well known that the Einstein tensor G for a Riemannian manifold defined by Gα^β = 1/2α^β ,Rα^β=g^α^β γ where Rγα and R are respectively the Ricci tensor and the scalar curvature of the manifold, plays an important part in Einstein s theory of gravitation as well as in proving some theorems in Riemannian geometry. In this work, we first obtain the generalized Einstein tensor for a Weyl manifold. Then, after studying some properties of generalized Einstein tensor, we prove that the conformal invariance of the generalized Einstein tensor implies the conformal invariance of the curvature tensor of the Weyl manifold and conversely. Moreover, we show that such Weyl manifolds admit a one-parameter family of hypersurfaces the orthogonal trajectories of which are geodesics. Finally, a necessary and sufficient condition in order that the generalized circles of a Weyl manifold be preserved by a conformal mapping is stated in terms of generalized Einstein tensors at corresponding points.展开更多
文摘In this article, after giving a necessary and sufficient condition for two Einstein- Weyl manifolds to be in conformal correspondence, we prove that any conformal mapping between such manifolds is generalized concircular if and only if the covector field of the conformal mapping is locally a gradient. Using this fact we deduce that any conformal mapping between two isotropic Weyl manifolds is a generalized concircular mapping. Moreover, it is shown that a generalized concircularly flat Weyl manifold is generalized concircular to an Einstein manifold and that its scalar curvature is prolonged covariant constant.
文摘It is well known that the Einstein tensor G for a Riemannian manifold defined by Gα^β = 1/2α^β ,Rα^β=g^α^β γ where Rγα and R are respectively the Ricci tensor and the scalar curvature of the manifold, plays an important part in Einstein s theory of gravitation as well as in proving some theorems in Riemannian geometry. In this work, we first obtain the generalized Einstein tensor for a Weyl manifold. Then, after studying some properties of generalized Einstein tensor, we prove that the conformal invariance of the generalized Einstein tensor implies the conformal invariance of the curvature tensor of the Weyl manifold and conversely. Moreover, we show that such Weyl manifolds admit a one-parameter family of hypersurfaces the orthogonal trajectories of which are geodesics. Finally, a necessary and sufficient condition in order that the generalized circles of a Weyl manifold be preserved by a conformal mapping is stated in terms of generalized Einstein tensors at corresponding points.
