To join an online seminar, connect to a Google Meet conference using the link above a few minutes prior to the beginning of the seminar.The work of lots of mathematicians is devoted to the study of manifolds of constant Ricci curvature, or Einstein manifolds. Recently, various generalizations of Einstein manifolds have been studied, one of which is (pseudo) Riemannian manifolds with zero Schouten-Weil tensor. In the case of constant scalar curvature, such manifolds are also known as Einstein-like manifolds, which were first considered by A. Gray, or as manifolds with zero divergence of the Weyl tensor. Note that in the case of four-dimensional Lie groups with a left-invariant pseudo-Riemannian metric, the class of manifolds with the trivial Schouten-Weyl tensor is not completely classified. For example, J. Calvaruso and A. Zaye classified pseudo-Riemannian Lie groups with trivial Weyl tensor, and also obtained a classification of Einstein and Ricci-parallel metrics on four-dimensional pseudo-Riemannian Lie groups.
In the dissertation, a classification of four-dimensional Lie groups with left-invariant pseudo-Riemannian metric and zero Schouten-Weyl tensor, which are neither conformally flat nor Ricci-parallel, is obtained, which completes the classification of four-dimensional locally homogeneous (pseudo) Riemannian manifolds with zero Schouten-Weyl tensor.
Another generalization of Einstein manifolds are Ricci solitons, new interest in which was initiated by G. Ya. Perelman's works on solving the hypothesis of A. Poincaré. An important tool in the study of solitons
Ricci are algebraic Ricci solitons on Lie groups, which were first considered by H. Laure.
The dissertation investigates algebraic Ricci solitons on n-dimensional metric Lie groups with the trivial Schouten-Weil tensor. A structural theorem on the structure of the Lie algebra of such solitons is proved, and possible forms of the Ricci operator are indicated. Nontrivial examples of algebraic Ricci solitons with a conformally flat pseudo-Riemannian metric are constructed.
In addition, a classification of four-dimensional locally homogeneous pseudo-Riemannian manifolds with a nontrivial stationary subgroup and an isotropic Schouten-Weil tensor is obtained, which continues the research begun earlier by V. V. Slavsky and E. D. Rodionov.
