On a Change in the Curvature of a Conformally Flat Metric under the Legendre Transformation
Abstract
It is known that the theory of conformally flat Riemannian metric is closely associated with pseudo-Euclidean geometry, due to the existence of the canonical isometric embedding conformally flat metric in pseudo-isotropic cone space. This fact was first noticed by H. Brinkmann, and later was used in the works of N. Kuyper. The geometry of homogeneous Riemannian manifolds with a conformally flat Riemannian metric was studied in the papers of A.D. Alekseevsky and B.N. Kimel’feld, in which their classification was given. In the inhomogeneous case such a classification does not exist, therefore, in the study of conformally flat Riemannian manifolds restrictions of various types are used: either on the dimension of the manifold, or on a topological structure, or on different types of Riemannian manifold curvatures of the conformally flat metric. In the last case, theorems on homogeneous Riemannian manifolds with a conformally flat metric of bounded one-dimensional curvature are well known, which were obtained by V.V. Slavsky and E.D. Rodionov. In this paper, we study the behavior of a one-dimensional curvature and the Ricci curvature under Legendre Transform of the conformally flat Riemannian metric. 10.14258/izvasu(2018)4-16Downloads
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References
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