Algebraic Ricci Solitons on Metric Lie Groups with Nondiagonalizable Ricci Operator

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.

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Homogeneous Ricci solitons on four-dimensional Lie groups with a left-invariant Riemannian metric

Doklady Mathematics ◽

10.1134/s1064562415060150 ◽

2015 ◽

Vol 92 (3) ◽

pp. 701-703 ◽

Cited By ~ 1

Author(s):

P. N. Klepikov ◽

D. N. Oskorbin ◽

E. D. Rodionov

Keyword(s):

Lie Groups ◽

Riemannian Metric ◽

Ricci Solitons ◽

Left Invariant Riemannian Metric

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Explicit Ricci Solitons on Nilpotent Lie Groups

Journal of Geometric Analysis ◽

10.1007/s12220-011-9237-5 ◽

2011 ◽

Vol 23 (1) ◽

pp. 47-72 ◽

Cited By ~ 2

Author(s):

Michael Bradford Williams

Keyword(s):

Lie Groups ◽

Nilpotent Lie Groups ◽

Ricci Solitons

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Homogeneous Ricci solitons

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2013-0044 ◽

2015 ◽

Vol 2015 (699) ◽

Cited By ~ 5

Author(s):

Michael Jablonski

Keyword(s):

Lie Groups ◽

Ricci Flow ◽

Isometry Group ◽

Ricci Soliton ◽

Ricci Solitons ◽

Solvable Lie Groups ◽

Compact Case ◽

Group Of Isometries ◽

Simply Connected ◽

Special Case

AbstractIn this work, we study metrics which are both homogeneous and Ricci soliton. If there exists a transitive solvable group of isometries on a Ricci soliton, we show that it is isometric to a solvsoliton. Moreover, unless the manifold is flat, it is necessarily simply-connected and diffeomorphic to ℝIn the general case, we prove that homogeneous Ricci solitons must be semi-algebraic Ricci solitons in the sense that they evolve under the Ricci flow by dilation and pullback by automorphisms of the isometry group. In the special case that there exists a transitive semi-simple group of isometries on a Ricci soliton, we show that such a space is in fact Einstein. In the compact case, we produce new proof that Ricci solitons are necessarily Einstein.Lastly, we characterize solvable Lie groups which admit Ricci soliton metrics.

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Pseudo-algebraic Ricci solitons on Einstein nilradicals

Advances in Geometry ◽

10.1515/advgeom-2020-0032 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Zaili Yan

Keyword(s):

Variational Method ◽

Lie Groups ◽

Lie Group ◽

Ricci Soliton ◽

Nilpotent Lie Groups ◽

Ricci Solitons ◽

Codimension One ◽

Abelian Ideal ◽

Einstein Nilradical

Abstract We develop a variational method to find pseudo-algebraic Ricci solitons on connected Lie groups. As applications, we prove that every Einstein nilradical admits a non-Riemannian algebraic Ricci soliton, and that any algebraic Ricci soliton on a semi-simple Lie group is Einstein. Furthermore, we construct several Lorentz algebraic Ricci solitons on the nilpotent Lie groups which have a codimension one abelian ideal.

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