Mathematical modeling in the study of semisymmetric connections on three-dimensional Lie groups with the metric of the Ricci soliton

Semisymmetric connections were first discovered by E. Cartan and are a natural generalization of the Levi-Civita connection. The properties of the parallel transfer of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano and other mathematicians. In this paper, a mathematical model is constructed for studying semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional unimodular Lie groups with left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that in this case there are nontrivial invariant semisimetric connections. Previously, the authors carried out similar studies in the class of Einstein metrics.

On Invariant Semisymmetric Connections on Three-Dimensional Non-Unimodular Lie Groups with the Metric of the Ricci Soliton

Metric connections with vector torsion, or semisymmetric connections, were first discovered by E. Cartan. They are a natural generalization of the Levi-Civita connection. The properties of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano, and other mathematicians. Ricci solitons are the solution to the Ricci flow and a natural generalization of Einstein's metrics. In the general case, they were investigated by many mathematicians, which was reflected in the reviews by H.-D. Cao, R.M. Aroyo — R. Lafuente. This question is best studied in the case of trivial Ricci solitons, or Einstein metrics, as well as the homogeneous Riemannian case. This paper investigates semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional non-unimodularLie groups with the left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that there are nontrivial invariant semisymmetric connections in this case. In addition, it is shown that there are nontrivial invariant Ricci solitons.

Eigenvalues of the Ricci Operator on Four-Dimensional Locally Homogeneous (Pseudo)Riemannian Manifolds with a Four-Dimensional Isotropy Subgroup

The problem of restoring a (pseudo)Riemannian manifold from a given Ricci operator was studied in the papers of many mathematicians. This problem was solved by O. Kowalski and S. Nikcevic for the case of three-dimensional locally homogeneous Riemannian manifolds. The work of G. Calvaruso and O. Kowalski contains the answer to the question above for the case of three –dimensional locally homogeneous Lorentzian manifolds. For the four-dimensional case, similar studies were carried out only in the case of Lie groups with a left-invariant Riemannian metric. The works of A.G. Kremlyov and Yu.G. Nikonorov presented the possible signatures of the eigenvalues of the Ricci operator. However, the question of recovering a four-dimensional Lie group with a left-invariant Riemannian metric from a given Ricci operator remains open. This paper is devoted to the study of the eigenvalues of the Ricci operator on four-dimensional locally homogeneous (pseudo)Riemannian manifolds with a four-dimensional isotropy subgroup. An algorithm for calculating the eigenvalues of the Ricci operator is presented. A theorem on the restoration of such manifolds from a given Ricci operator is proved. It is established that such possibility can happen only in the case when the prescribed operator is diagonalizable and has a unique eigenvalue of multiplicity four.

Randers Ricci soliton homogeneous nilmanifolds

Let $F$ be a left-invariant Randers metric on a simply connected nilpotent Lie group $N$, induced by a left-invariant Riemannian metric $\hat{\boldsymbol{a}}$ and a vector field $X$ which is $I_{\hat{\boldsymbol{a}}}(M)$-invariant. We show that if the Ricci flow equation has a unique solution then, $(N,F)$ is a Ricci soliton if and only if $(N,F)$ is a semialgebraic Ricci soliton.

Classification of Douglas (α,β)-metrics on five-dimensional nilpotent Lie groups

In this paper, we classify all simply connected five-dimensional nilpotent Lie groups which admit [Formula: see text]-metrics of Berwald and Douglas type defined by a left invariant Riemannian metric and a left invariant vector field. During this classification, we give the geodesic vectors, Levi-Civita connection, curvature tensor, sectional curvature and [Formula: see text]-curvature.

Left invariant Randers metrics of Berwald type on tangent Lie groups

Let [Formula: see text] be a Lie group equipped with a left invariant Randers metric of Berward type [Formula: see text], with underlying left invariant Riemannian metric [Formula: see text]. Suppose that [Formula: see text] and [Formula: see text] are lifted Randers and Riemannian metrics arising from [Formula: see text] and [Formula: see text] on the tangent Lie group [Formula: see text] by vertical and complete lifts. In this paper, we study the relations between the flag curvature of the Randers manifold [Formula: see text] and the sectional curvature of the Riemannian manifold [Formula: see text] when [Formula: see text] is of Berwald type. Then we give all simply connected three-dimensional Lie groups such that their tangent bundles admit Randers metrics of Berwarld type and their geodesics vectors.

Three-dimensional solvsolitons and the minimality of the corresponding submanifolds

In this paper, we define the corresponding submanifolds to left-invariant Riemannian metrics on Lie groups, and study the following question: does a distinguished left-invariant Riemannian metric on a Lie group correspond to a distinguished submanifold? As a result, we prove that the solvsolitons on three-dimensional simply-connected solvable Lie groups are completely characterized by the minimality of the corresponding submanifolds.