Generalized Einstein’s tensor for a Weyl manifold and its applications
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CitationÖzdeğer, A. (2009). Generalized Einstein’s tensor for a Weyl manifold and its applications. Maltepe Üniversitesi. s. 64.
A differentiable manifold having a torsion-free connection ∇ and a conformal class C[g] of metrics which is preserved by ∇ is called a Weyl manifold. The condition involved in this definition can be expressed as ∇g = 2(g ⊗ w) for some 1-form w  . It is well known that Einstein’s tensor G for a Riemannian manifold defined by Gβ α = R β α − 1 2 δ β αR, Rβ α = g βγRαγ where R β α and R 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 basic theorems in Riemannian geometry . In this work , we obtain the generalized Einstein’s tensor for Weyl manifolds by using the second Bianchi identity for such manifolds obtained in  . Then, we deduce the following results : (a) Any 2-dimensional Einstein-Weyl manifold has a vanishing generalized Einstein’s tensor, (b) A Weyl manifold and its Liouville transformation have the same generalized Einstein’s tensor, (c) If the 1-form w for an Einstein-Weyl manifold is locally a gradient, then the scalar curvature of the manifold is prolonged covariant constant.
SourceInternational Conference of Mathematical Sciences
- Makale Koleksiyonu 
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