Aboute Symmetric Ricci Flow On the Three-Dimensional Heisenberg Group

The Ricci flow equation was first studied by R. Hamilton for the Levi-Civita connection. It plays an important role in Riemannian geometry. The class of semi-symmetric connections, described first by E. Cartan, contains the Levi-Civita connection. Thus, it is natural to consider the Ricci flow on Riemannian manifolds with a semi-symmetric connection. It is known that the Ricci tensor of a semi-symmetric connection is, generally speaking, not symmetric. Therefore, it is necessary to consider the symmetric part of the Ricci tensor and the symmetric Ricci flow with respect to this tensor. This paper studies the symmetric Ricci flow on three-dimensional unimodular Lie groups with a semi-symmetric connection. The flow equation in the J. Milnor coordinate system is reduced to a system of algebraic and differential equations. The symmetric Ricci flow on a three-dimensional unimodular group with the J. Milnor metric with respect to a semi-symmetric connection is found by sequentially solving the subsystem of algebraic equations and then substituting the obtained solution into the system of differential equations. The three-dimensional Heisenberg group is considered as a test example.