Özdeğer, Abdülkadir2024-07-122024-07-122009Özdeğer, A. (2009). Generalized Einstein’s tensor for a Weyl manifold and its applications. Maltepe Üniversitesi. s. 64.9.78605E+12https://www.maltepe.edu.tr/Content/Media/CkEditor/03012019014112056-AbstractBookICMS2009Istanbul.pdf#page=76https://hdl.handle.net/20.500.12415/2393A 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 [1] . It is well known that Einstein’s tensor G for a Riemannian manifold defined by G? ? = R ? ? ? 1 2 ? ? ?R, R? ? = g ??R?? where R ? ? 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 [2]. In this work , we obtain the generalized Einstein’s tensor for Weyl manifolds by using the second Bianchi identity for such manifolds obtained in [3] . 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.enCC0 1.0 Universalinfo:eu-repo/semantics/openAccessConference Object6564