Symmetric Ricci Flows of Semisymmetric Connections on Three-Dimensional Metrical Lie Groups: An Analysis

Abstract

The study of Ricci flows, which describe the deformation of (pseudo) Riemannian metrics on a manifold, and their solutions, Ricci solitons, has garnered much attention from mathematicians. However, previous studies have typically focused on manifolds with Levi-Civita connections. This paper breaks new ground by considering manifolds with semisymmetric connections, which also include the Levi-Civita connection. Metric connections with vector torsion, or semisymmetric connections, were first studied by E. Cartan on (pseudo) Riemannian manifolds. Later, K. Yano and I. Agricola studied tensor fields and geodesic lines of such connections, while P.N. Klepikov,
 E.D. Rodionov, and O.P. Khromova considered the Einstein equation of semisymmetric connections on three-dimensional locally homogeneous (pseudo) Riemannian manifolds. Because the Ricci tensor of a semisymmetric connection is not symmetric in general, we focus on studying the symmetric and skew-symmetric parts of the Ricci tensor. Specifically, we investigate symmetric Ricci flows on three-dimensional Lie groups with J. Milnor's left-invariant (pseudo) Riemannian metric and E. Cartan's semisymmetric connection.