Homogeneous Ricci Solitons Research Articles

On an Equation in the Ricci Solitons Theory with a Semisymmetric Connection

The study of Ricci solitons and invariant Ricci solitons with connections of various types has garnered much attention from many mathematicians. 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.
 In the previous paper, the authors studied invariant Ricci solitons with a semisymmetric connection. They are an important subclass of the class of homogeneous Ricci solitons. We obtained the classification of invariant Ricci solitons on three-dimensional Lie groups with a left-invariant Riemannian metric and a semisymmetric connection different from the Levi-Civita connection. Also, the existence of invariant Ricci solitons with a non-conformal Killing vector field was proved for the such case. Moreover, a part of the proofs was obtained using the analytical calculation software packages.
 In this paper, we investigate invariant Ricci solitons on three-dimensional nonunimodular Lie groups with a left-invariant Riemannian metric and a semisymmet-ric connection. Analytical proofs of all theorems completing the classification of such solitons are presented.

Generalized Ricci Solitons on Non-reductive Four-Dimensional Homogeneous Spaces

In the present paper, we consider the non-reductive four-dimensional homogeneous spaces and we classify homogeneous generalized Ricci solitons on these spaces. We show that any non-reductive four-dimensional homogeneous space admits the least in a generalized Ricci soliton. Also, we will prove that non-reductive four-dimensional homogeneous spaces have non-trivial Killing vector fields and these spaces exclusive of types A1, A4 and B2 are Einstein manifold and admit in non-trivial homogeneous Ricci solitons.

Siklos spacetimes as homogeneous Ricci solitons

We consider a well-known class of homogeneous solutions of the Einstein–Maxwell equations found by Siklos and prove that all these spacetimes are nontrivial Ricci solitons. In particular, these examples show that several rigidity results for homogeneous Lorentzian gradient Ricci solitons do not extend to the non-gradient case.

Laplacian flow of homogeneous G2-structures and its solitons

We use the bracket flow/algebraic soliton approach to study the Laplacian flow of $G_2$-structures and its solitons in the homogeneous case. We prove that any homogeneous Laplacian soliton is equivalent to a semi-algebraic soliton (i.e.\ a $G$-invariant $G_2$-structure on a homogeneous space $G/K$ that flows by pull-back of automorphisms of $G$ up to scaling). Algebraic solitons are geometrically characterized among Laplacian solitons as those with a `diagonal' evolution. Unlike the Ricci flow case, where any homogeneous Ricci soliton is isometric to an algebraic soliton, we have found, as an application of the above characterization, an example of a left-invariant closed semi-algebraic soliton on a nilpotent Lie group which is not equivalent to any algebraic soliton. The (normalized) bracket flow evolution of such a soliton is periodic. In the context of solvable Lie groups with a codimension-one abelian normal subgroup, we obtain long time existence for any closed Laplacian flow solution; furthermore, the norm of the torsion is strictly decreasing and converges to zero. We also classify algebraic solitons in this class and exhibit several explicit examples of closed expanding Laplacian solitons.

Алгебраические солитоны Риччи на метрических группах Ли с недиагонализируемым оператором Риччи

In recent years, various generalizations of Einstein manifolds are actively studied, for example, manifolds with the trivial Schouten-Weyl tensor, and Ricci solitons, which were first considered by R. Hamilton. Ricci solitons on homogeneous (pseudo)Rieman-nian spaces and, in particular, on the Lie groups have been studied by many mathematicians. For example, there are no nontrivial homogeneous invariant Ricci solitons on three and four-dimensional Lie groups with a left-invariant Riemannian metric. A similar result was proved for the unimodular Lie groups with a left-invariant Riemannian metric in any dimension. However, this question is still an open problem for nonunimodular Lie groups of dimension more than 4. Another important example of Ricci solitons is the case of algebraic Ricci solitons on Lie groups, first considered by J. Lauret. Later, it was proved that every algebraic Ricci soliton on a Lie group with left-invariant (pseudo)Riemannian metric is a homogeneous Ricci soliton. This paper shows the existence of non-trivial algebraic and homogeneous invariant Ricci solitons on conformally flat Lie groups in the case of nondiagonalizable Ricci operator. Also, a non-diagonalizable Ricci operator on Lie groups with harmonic Weyl tensor is demonstrated.

Homogeneous Ricci soliton hypersurfaces in the complex hyperbolic spaces

A Lie hypersurface in the complex hyperbolic space is an orbit of a cohomogeneity one action without singular orbit. In this paper, we classify Ricci soliton Lie hypersurfaces in the complex hyperbolic spaces.

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

Homogeneous Ricci solitons on four-dimensional Lie groups with a left-invariant Riemannian metric are studied. The absence of nontrivial homogeneous invariant Ricci solitons is proved. The algebraic soliton equations are solved in terms of the structure constants of the metric Lie algebra.

Conformally Flat Pseudo-riemannian Homogeneous Ricci Solitons 4-spaces

We consider four dimensional conformally flat homogeneous pseudo Riemannian manifolds. According to forms (Seger types) of the Ricci operator, we provide a full classification of four dimensional pseudo Riemannian conformally flat homogeneous Ricci solitons.

Four-dimensional pseudo-Riemannian homogeneous Ricci solitons

We consider four-dimensional homogeneous pseudo-Riemannian manifolds with non-trivial isotropy and completely classify the cases giving rise to non-trivial homogeneous Ricci solitons. In particular, we show the existence of non-compact homogeneous (and also invariant) pseudo-Riemannian Ricci solitons which are not isometric to solvmanifolds, and of conformally flat homogeneous pseudo-Riemannian Ricci solitons which are not symmetric.

Strongly solvable spaces

This work builds on the foundation laid by Gordon and Wilson in the study of isometry groups of solvmanifolds, i.e. Riemannian manifolds admitting a transitive solvable group of isometries. We restrict ourselves to a natural class of solvable Lie groups called almost completely solvable; this class includes the completely solvable Lie groups. When the commutator subalgebra contains the center, we have a complete description of the isometry group of any left-invariant metric using only metric Lie algebra information. Using our work on the isometry group of such spaces, we study quotients of solvmanifolds. Our first application is to the classification of homogeneous Ricci soliton metrics. We show that the verification of the Generalized Alekseevsky Conjecture reduces to the simply-connected case. Our second application is a generalization of a result of Heintze on the rigidity of existence of compact quotients for certain homogeneous spaces. Heintze's result applies to spaces with negative curvature. We remove all the geometric requirements, replacing them with algebraic requirements on the homogeneous structure.

Однородные инвариантные солитоны Риччи на четырехмерных группах Ли

Ricci solitons are important generalizations of Einstein metrics on Riemann manifolds. These metrics were first investigated by Hamilton. Ricci solitons are relevant to the solutions of the Ricci flow. Homogeneous Riemannian metric on the homogeneous space G/H satisfying the Ricci soliton is called the homogeneous Ricci soliton. Such metrics have been studied by many mathematicians. The classification of homogeneous Ricci solitons is known in small dimensions only, and it is not exhaustive. It is known that for three-dimensional Lie groups with left-invariant Riemannian metric Ricci soliton equation has no solution in the class of left-invariant vector fields. A similar fact is proved for unimodular Lie groups with left-invariant Riemannian metric of any finite dimension. However, the existence problem for non-trivial invariant Ricci solitons on nonunimodular Lie groups of dimension > 3 remains open. In this paper, we obtain the solution of this problem in dimension 4. The soliton equation by generalized Milnor’s frames reduced to the system of polynomial equations. The absence of nontrivial homogeneous invariant Ricci solitons on fourdimensional Lie groups is proved.DOI 10.14258/izvasu(2015)1.2-21

On homogeneous warped product Einstein metrics

In this article we study homogeneous warped product Einstein metrics and its connections with homogeneous Ricci solitons. We show that homogeneous $(\lambda,n+m)$-Einstein manifolds (which are the bases of homogeneous warped product Einstein metrics) are one-dimensional extensions of algebraic solitons. This answers a question from a paper of C. He, P. Petersen and W. Wylie, where they prove the converse statement. Our proof is strongly based on their results, but it also makes use of sharp tools from the theory of homogeneous Ricci solitons. As an application, we obtain that any homogeneous warped product Einstein metric with homogeneous base is diffeomorphic to a product of homogeneous Einstein manifolds.

Homogeneous Ricci solitons are algebraic

In this short note, we show that homogeneous Ricci solitons are algebraic. As an application, we see that the generalized Alekseevskii conjecture is equivalent to the Alekseevskii conjecture.

Explicit description of three-dimensional homogeneous Ricci solitons

Explicit description of three-dimensional homogeneous Ricci solitons

Generic properties of homogeneous Ricci solitons

Abstract We discuss the geometry of homogeneous Ricci solitons. After showing the nonexistence of compact homogeneous and non-compact steady homogeneous solitons, we concentrate on the study of left invariant Ricci solitons. We show that, in the unimodular case, the Ricci soliton equation does not admit solutions in the set of left invariant vector fields. We prove that a left invariant soliton of gradient type must be a Riemannian product with non-trivial Euclidean de Rham factor. As an application of our results we prove that any generalized metric Heisenberg Lie group is a non-gradient left invariant Ricci soliton of expanding type.

Warped product Einstein metrics on homogeneous spaces and homogeneous Ricci solitons

Abstract In this paper we consider connections between Ricci solitons and Einstein metrics on homogeneous spaces. We show that a semi-algebraic Ricci soliton admits an Einstein one-dimensional extension if the soliton derivation can be chosen to be normal. Using our previous work on warped product Einstein metrics, we show that every normal semi-algebraic Ricci soliton also admits a k-dimensional Einstein extension for any k ≥ 2. We also prove converse theorems for these constructions and some geometric and topological structure results for homogeneous warped product Einstein metrics. In the appendix we give an alternative approach to semi-algebraic Ricci solitons which naturally leads to a definition of semi-algebraic Ricci solitons in the non-homogeneous setting.

Homogeneous Ricci solitons

Abstract In 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 ℝ n . In 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.

ON HOMOGENEOUS RICCI SOLITONS

We study the evolution of homogeneous Ricci solitons under the bracket flow, a dynamical system on the space of all homogeneous spaces of dimension n with a q-dimensional isotropy, which is equivalent to the Ricci flow for homogeneous manifolds. We prove that algebraic solitons (i.e. the Ricci operator is a multiple of the identity plus a derivation) are precisely the fixed points of the system, and that a homogeneous Ricci soliton is isometric to an algebraic soliton if and only if the corresponding bracket flow solution is not chaotic, in the sense that its omega-limit set consists of a single point. We also geometrically characterize algebraic solitons among homogeneous Ricci solitons as those for which the Ricci flow solution is simultaneously diagonalizable.

Ricci Solitons and Geometry of Four-dimensional Non-reductive Homogeneous Spaces

Abstract We study the geometry of non-reductive 4-dimensional homogeneous spaces. In particular, after describing their Levi-Civita connection and curvature properties, we classify homogeneous Ricci solitons on these spaces, proving the existence of shrinking, expanding and steady examples. For all the non-trivial examples we find, the Ricci operator is diagonalizable.

Three-dimensional Lorentzian homogeneous Ricci solitons

We study three-dimensional Lorentzian homogeneous Ricci solitons, proving the existence of shrinking, expanding and steady Ricci solitons. For all the non-trivial examples, the Ricci operator is not diagonalizable and has three equal eigenvalues.