Algebraic Ricci Solitons Research Articles

Affine Algebraic Ricci Solitons Associated to the Yano Connections on Three-Dimensional Lorentzian Lie Groups

In this paper, we compute curvatures of Yano connections on three-dimensional Lorentzian Lie groups with some product structure. We define affine algebraic Ricci solitons associated to Yano connections and classify left-invariant affine algebraic Ricci solitons associated to Yano connections on three-dimensional Lorentzian Lie groups.

Canonical Connections and Algebraic Ricci Solitons of Three-dimensional Lorentzian Lie Groups

In this paper, the author computes canonical connections and Kobayashi-Nomizu connections and their curvature on three-dimensional Lorentzian Lie groups with some product structure. He defines algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections. He classifies algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.

Double Extensions on Riemannian Ricci Nilsolitons

In this paper, we study left invariant Lorentz algebraic Ricci solitons on nilpotent Lie groups. Based on the concept of Lorentz datum and the technique of double extensions, we are able to construct Lorentz Ricci nilsolitons from any Riemannian Ricci nilsoliton. Conversely, we prove that any Lorentz Ricci nilsoliton with degenerate center is a double extension of a Riemannian Ricci nilsoliton with respect to a Lorentz data. Moreover, we provide a strategy to classify Lorentz Ricci nilsolitons with degenerate center. As an application, we present a large number of Lorentz Ricci nilsolitons based on Abelian Riemannian Ricci solitons. Finally, a family of left invariant Lorentz Ricci nilsolitons on a 9-dimensional nilpotent Lie algebra is given.

Mathematical Modeling in Problems of Homogeneous (Pseudo)Riemaimian Geometry

Currently, mathematical and computer modeling, as well as systems of symbolic calculations, are actively used in many areas of mathematics. Popular computer math systems as Maple, Mathematica, MathCad, MatLab allow not only to perform calculations using symbolic expressions but also solve algebraic and differential equations (numerically and analytically) and visualize the results. Differential geometry, like other areas of modern mathematics, uses new computer technologies to solve its own problems. The applying is not limited only to numerical calculations; more and more often, computer mathematics systems are used for analytical calculations. At the moment, there are many examples that prove the effectiveness of systems of analytical calculations in the proof of theorems of differential geometry.This paper demonstrates how symbolic computation packages can be used to classify neither conformally flat nor Ricci parallel four-dimensional Lie groups with leftinvariant (pseudo)Riemannian metric of the algebraic Ricci soliton with the zero Schouten-Weyl tensor.

On the geometrical properties of Heisenberg groups

In [20] the existence of major differences about totally geodesic two-dimensional foliations between Riemannian and Lorentzian geometry of the Heisenberg group $H_{3}$ is proved. Our aim in this paper is to obtain a comparison on some other geometrical properties of these spaces. Interesting behaviours are found. Also the non-existence of left-invariant Ricci and Yamabe solitons and the existence of algebraic Ricci soliton in both Riemannian and Lorentzian cases are proved. Moreover, all of the descriptions of their homogeneous Riemannian and Lorentzian structures and their types are obtained. Besides, all the left-invariant generalized Ricci solitons and unit time-like vector fields which are spatially harmonic are completely determined.

On the Geometry of Higher Dimensional Heisenberg Groups

In this paper, we first completely determine all left-invariant generalized Ricci solitons on the Heisenberg group \(H_{2n+1}\) equipped with any left-invariant Riemannian and Lorentzian metric that this Lie group admits. Then, we explicitly calculate the energy of an arbitrary left-invariant vector field V on these spaces and in the Lorentzian cases we determine the exact form of all left-invariant unit time-like vector fields which are spatially harmonic. We also obtain all of the descriptions of their homogeneous Riemannian and Lorentzian structures and explicitly distinguish their types. Finally, we investigate parallel hypersurfaces of these spaces and show that these spaces never admit any totally geodesic hypersurface. The existence of algebraic Ricci solitons and the non-existence of left-invariant Ricci solitons and Yamabe solitons on these spaces in both Riemannian and Lorentzian cases is proved. Also, different behaviors regarding the existence of harmonic maps, critical points for the energy functional restricted to vector fields and some equations in Riemannian and Lorentzian cases are found.

Conformally Flat Algebraic Ricci Solitons on Lie Groups

The paper is devoted to the study of conformally flat Lie groups with left-invariant (pseudo) Riemannianmetric of an algebraic Ricci soliton. Previously conformally flat algebraic Ricci solitons on Lie groups have been studied in the case of small dimension and under an additional diagonalizability condition on the Ricci operator. The present paper continues these studies without the additional requirement that the Ricci operator be diagonalizable. It is proved that any nontrivial conformally flat algebraic Ricci soliton on a Lie group must be steady and have Ricci operator of Segre type {(1... 1 2)} with a unique eigenvalue (equal to 0).

On the geometry of para-hypercomplex 4-dimensional Lie groups

In this paper we first completely determine left-invariant generalized Ricci solitons and Einstein-like metrics on para-hypercomplex 4- dimensional Lie groups equipped with a left-invariant Riemannian metric g and a left-invariant Lorentzian metric g ̂ 1 . Then we obtain the exact form of all harmonic maps and prove that contrary to the Lorentzian case, there exist spaces where their energy functional restricted to vector fields of the same length admit any left-invariant vector fields X as critical points. We also obtain the spaces on which all homogeneous Riemannian structures coincide with all homogeneous Lorentzian structures. Finally, we give the complete classification and explicitly describe totally geodesic hypersurfaces of these spaces in both Riemannian and Lorentzian cases. The existence of algebraic Ricci solitons which are neither Einstein nor Ricci and Yamabe solitons is proved. Also remarkable differences between the existence of Einstein, Einstein-like metrics, locally symmetric, conformal flatness and some equations in Riemannian and Lorentzian cases are given.

Conformally Flat Algebraic Ricci Solitons on Lie Groups

Данная работа посвящена изучению конформно плоских групп Ли с левоинвариантной (псевдо)римановой метрикой алгебраического солитона Риччи. Ранее конформно плоские алгебраические солитоны Риччи на группах Ли изучались в случае малой размерности, а также при дополнительном условии диагонализируемости оператора Риччи. Настоящая работа продолжает данные исследования без дополнительного требования диагонализируемости оператора Риччи. Получена теорема о том, что любой нетривиальный конформно плоский алгебраический солитон Риччи на группе Ли обязан быть устойчивым и иметь оператор Риччи с типом Сегре $\{(1 …1 2)\}$ и единственным собственным значением, равным нулю. Библиография: 21 название.

Algebraic Ricci solitons on metric Lie groups with zero Schouten–Weyl tensor

Algebraic Ricci solitons on Lie groups with left-invariant (pseudo)Riemannian metric and zero Schouten–Weyl tensor are studied. The absence of nontrivial algebraic Ricci solitons on metric Lie groups with zero Schouten–Weyl tensor and diagonalizable Ricci operator is proved.

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.

Применение пакетов символьных вычислений к исследованию алгебраических солитонов Риччи на однородных (псевдо)римановых многообразиях

Применение пакетов символьных вычислений к исследованию алгебраических солитонов Риччи на однородных (псевдо)римановых многообразиях

Algebraic Ricci solitons of three-dimensional Lorentzian Lie groups

We study algebraic Ricci solitons of three-dimensional Lorentzian Lie groups. All algebraic Ricci solitons that we obtain are solvsolitons. In particular, we obtain new solitons on G2, G5, and G6, and we prove that, contrary to the Riemannian case, Lorentzian Ricci solitons need not be algebraic Ricci solitons.

On Lorentzian Ricci solitons on nilpotent Lie groups

The aim of this article is to exhibit the variety of different Ricci soliton structures that a nilpotent Lie group can support when one allows for the metric tensor to be Lorentzian. In stark contrast to the Riemannian case, we show that a nilpotent Lie group can support a number of non-isometric Lorentzian Ricci soliton structures with decidedly different qualitative behaviors and that Lorentzian Ricci solitons need not be algebraic Ricci solitons. The analysis is carried out by classifying all left invariant Lorentzian metrics on the connected, simply-connected five-dimensional Lie group having a Lie algebra with basis vectors E1,E2,E3,E4 and E5 and non-trivial bracket relations E1,E5=E3 and E2,E5=E4, investigating the various curvature properties of the resulting families of metrics, and classifying all Lorentzian Ricci soliton structures.

Ricci solitons on low-dimensional generalized symmetric spaces

We consider three- and four-dimensional pseudo-Riemannian generalized symmetric spaces, whose invariant metrics were explicitly described in [15]. While four-dimensional pseudo-Riemannian generalized symmetric spaces of types A, C and D are algebraic Ricci solitons, the ones of type B are not so. The Ricci soliton equation for their metrics yields a system of partial differential equations. Solving such system, we prove that almost all the four-dimensional pseudo-Riemannian generalized symmetric spaces of type B are Ricci solitons. These examples show some deep differences arising for the Ricci soliton equation between the Riemannian and the pseudo-Riemannian cases, as any homogeneous Riemannian Ricci soliton is algebraic [21]. We also investigate three-dimensional generalized symmetric spaces of any signature and prove that they are Ricci solitons.

Examples of Algebraic Ricci Solitons in the Pseudo-Riemannian Case

This paper provides a study of algebraic Ricci solitons in the pseudo-Riemannian case. In the Riemannian case, all nontrivial homogeneous algebraic Ricci solitons are expanding algebraic Ricci solitons. We obtain a steady algebraic Ricci soliton and a shrinking algebraic Ricci soliton in the Lorentzian setting.

Linear stability of algebraic Ricci solitons

Abstract We consider a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics, and we study the linear stability of those solutions relative to the flow. After deriving various criteria that imply linear stability, we turn our attention to left-invariant soliton metrics on (non-compact) simply connected solvable Lie groups and prove linear stability of many such metrics. These include an open set of two-step solvsolitons, all two-step nilsolitons, two infinite families of three-step solvable Einstein metrics, all nilsolitons of dimensions six or less, and all solvable Einstein metrics of dimension seven or less with codimension-one nilradical. For each linearly stable metric, dynamical stability follows from a generalization of the techniques of Guenther, Isenberg, and Knopf.

Dynamical stability of algebraic Ricci solitons

Abstract We consider dynamical stability for a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics. Our focus is on homogeneous metrics on non-compact manifolds. Following the program of Guenther, Isenberg, and Knopf, we define a class of weighted little Hölder spaces with certain interpolation properties that allow the use of maximal regularity theory and the application of a stability theorem of Simonett. With this, we derive two stability theorems, one for a class of Einstein metrics and one for a class of non-Einstein Ricci solitons. Using linear stability results of Jablonski, Petersen, and the first author, we obtain dynamical stability for many specific Einstein and Ricci soliton metrics on simply connected solvable Lie groups.

On the Isometry Groups of Invariant Lorentzian Metrics on the Heisenberg Group

This work concerns the invariant Lorentzian metrics on the Heisenberg Lie group of dimension three \({{\rm H}_3(\mathbb{R})}\) and the bi-invariant metrics on the solvable Lie groups of dimension four. We start with the indecomposable Lie groups of dimension four admitting bi-invariant metrics and which act on \({{\rm H}_3(\mathbb{R})}\) by isometries and we study some geometrical features on these spaces. On \({{\rm H}_3(\mathbb{R})}\) , we prove that the property of the metric being proper naturally reductive is equivalent to the property of the center being non-degenerate. These metrics are Lorentzian algebraic Ricci solitons.

Four-Dimensional Pseudo-Riemannian Generalized Symmetric Spaces Which are Algebraic Ricci Solitons

We classify, up to isometry, non-symmetric simply-connected four-dimensional pseudo-Riemannian generalized symmetric spaces which are algebraic Ricci solitons. It turns out that those of Cerny–Kowalski’s types A, C and D are algebraic Ricci solitons, whereas those of type B are not. Thus, we give new examples of algebraic Ricci solitons.