Three-dimensional Lie Groups Research Articles

Locally Homogeneous Pseudo-Riemannian 4-Manifolds with Isotropic Weyl Tensor

Papers of many mathematicians are devoted to studying of conformally flat (i.e., with the trivial Weyl tensor) (pseudo)Riemannian manifolds. Moreover, one can consider manifolds with Weyl tensors having zero squared length while itself being non zero. Also, such manifolds are called manifolds with isotropic Weyl tensors.In the case of Riemannian metric, the squared length of the tensor in some orthonormal basis is the sum of the squares of all components. The squared length equals to zero if the tensor itself is trivial. Therefore, it is natural to consider only the pseudo-Riemannian metric case. For the dimension of 3, the Weyl tensor is trivial, and the Schouten-Weyl tensor (also known as the Cotton tensor) is identical with the Weyl tensor. In the paper of Rodionov E.D., Slavskii V.V., Chibrikova L.N., the Schouten-Weyl tensor was investigated for a left-invariant Lorentzian metric on three-dimensional Lie groups, including the problem of its isotropy.In this paper, results of the study of fourdimensional locally homogeneous spaces with nontrivial isotropy subgroup and with invariant pseudo-Riemannian metric and an isotropic Weyl tensor are presented.DOI 10.14258/izvasu(2018)1-17

Numerical continuation of invariant solutions of the complex Ginzburg–Landau equation

We consider the problem of computation and deformation of group orbits of solutions of the complex Ginzburg–Landau equation (CGLE) with cubic nonlinearity in 1+1 space–time dimension invariant under the action of the three-dimensional Lie group of symmetries A(x,t)→eiθA(x+σ,t+τ). From an initial set of group orbits of invariant solutions, for a particular point in the parameter space of the CGLE, we obtain new sets of group orbits of invariant solutions via numerical continuation along paths in the moduli space. The computed solutions along the continuation paths are unstable, and have multiple modes and frequencies active in their spatial and temporal spectra, respectively. Structural changes in the moduli space resulting in symmetry gaining / breaking associated often with the spatial reflection symmetry A(x,t)→A(−x,t) of the CGLE were frequently uncovered in the parameter regions traversed.

Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups

Abstract We investigate the isometry groups of the left-invariant Riemannian and sub-Riemannian structures on simply connected three-dimensional Lie groups. More specifically, we determine the isometry group for each normalized structure and hence characterize for exactly which structures (and groups) the isotropy subgroup of the identity is contained in the group of automorphisms of the Lie group. It turns out (in both the Riemannian and sub-Riemannian cases) that for most structures any isometry is the composition of a left translation and a Lie group automorphism.

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 Homogeneous Generalized Ricci Solitons

We study three-dimensional generalized Ricci solitons, both in Riemannian and Lorentzian settings. We shall determine their homogeneous models, classifying left-invariant generalized Ricci solitons on three-dimensional Lie groups.

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.

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 algebraic solitons for geometric evolution equations on three-dimensional Lie groups

The relationship between algebraic soliton metrics and self-similar solutions of geometric evolution equations on Lie groups is investigated. After discussing the general relationship between algebraic soliton metrics and self-similar solutions to geometric evolution equations, we investigate the cross curvature flow and the second order renormalization group flow on simply-connected, three-dimensional, unimodular Lie groups, providing a complete classification of left invariant algebraic solitons that give rise to self-similar solutions of the corresponding flows on such spaces.

Invariant nonholonomic Riemannian structures on three-dimensional Lie groups

We consider Riemannian manifolds endowed with a nonholonomic distribution. These structures model mechanical systems with a (positive definite) quadratic Lagrangian and nonholonomic constraints linear in velocities. We classify the left-invariant nonholonomic Riemannian structures on three-dimensional simply connected Lie groups, and describe the equivalence classes in terms of some basic isometric invariants. The classification naturally splits into two cases. In the first case, it reduces to a classification of left-invariant sub-Riemannian structures. In the second case, we find a canonical frame with which to directly compare equivalence classes.

Milnor-type theorems for left-invariant Riemannian metrics on Lie groups

For all left-invariant Riemannian metrics on three-dimensional unimodular Lie groups, there exist particular left-invariant orthonormal frames, so-called Milnor frames. In this paper, for any left-invariant Riemannian metrics on any Lie groups, we give a procedure to obtain an analogous of Milnor frames, in the sense that the bracket relations among them can be written with relatively smaller number of parameters. Our procedure is based on the moduli space of left-invariant Riemannian metrics. Some explicit examples of such frames and applications will also be given.

Classiffications of special curves in the three-dimensional Lie group

Classiffications of special curves in the three-dimensional Lie group

Sectional and Ricci Curvature for Three-Dimensional Lie Groups

Formulas for the Riemann and Ricci curvature tensors of an invariant metric on a Lie group are determined. The results are applied to a systematic study of the curvature properties of invariant metrics on three-dimensional Lie groups. In each case the metric is reduced by using the automorphism group of the associated Lie algebra. In particular, the maximum and minimum values of the sectional curvature function are determined.

Graph surfaces over three-dimensional Lie groups with sub-Riemannian structure

We study some class of graph mappings on certain three-dimensional Lie groups, derive special differential properties of these mappings, and prove the area formula for graph surfaces.

Superintegrability of left-invariant sub-Riemannian structures on unimodular three-dimensional Lie groups

We consider left-invariant sub-Riemannian problems on three-dimensional unimodular Lie groups. We show that the Hamiltonian system of the Pontryagin maximum principle for such problems is Liouville integrable and even superintegrable (i.e., has four independent integrals, three of which are in involution).

The subelliptic heat kernel on the three-dimensional solvable Lie groups

Abstract We study the subelliptic heat kernels of the CR three-dimensional solvable Lie groups. We first classify all left-invariant sub-Riemannian structures on three-dimensional solvable Lie groups and obtain representations of these groups. We give expressions for the heat kernels on these groups and obtain heat semigroup gradient bounds using a new type of curvature-dimension inequality.

Invariant Ricci collineations on three-dimensional Lie groups

We determine all left-invariant Ricci collineations on three-dimensional Lie groups.

Diffeomorphism Spline

Conventional splines offer powerful means for modeling surfaces and volumes in three-dimensional Euclidean space. A one-dimensional quaternion spline has been applied for animation purpose, where the splines are defined to model a one-dimensional submanifold in the three-dimensional Lie group. Given two surfaces, all of the diffeomorphisms between them form an infinite dimensional manifold, the so-called diffeomorphism space. In this work, we propose a novel scheme to model finite dimensional submanifolds in the diffeomorphism space by generalizing conventional splines. According to quasiconformal geometry theorem, each diffeomorphism determines a Beltrami differential on the source surface. Inversely, the diffeomorphism is determined by its Beltrami differential with normalization conditions. Therefore, the diffeomorphism space has one-to-one correspondence to the space of a special differential form. The convex combination of Beltrami differentials is still a Beltrami differential. Therefore, the conventional spline scheme can be generalized to the Beltrami differential space and, consequently, to the diffeomorphism space. Our experiments demonstrate the efficiency and efficacy of diffeomorphism splines. The diffeomorphism spline has many potential applications, such as surface registration, tracking and animation.

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

Complete Left-Invariant Affine Structures on Solvable Non-Unimodular Three-Dimensional Lie Groups

In this paper, we shall use a method based on the theory of extensions of left-symmetric algebras to classify complete left-invariant affine real structures on solvable non-unimodular three-dimensional Lie groups.

On the degeneration of some 3-manifold geometries via unit groups of quaternion algebras

A two parameter continuous family of three-dimensional Lie groups with a left invariant Riemannian metric is defined. Each of these Lie groups is the unit group of a quaternion algebra. All the possible left invariant Riemannian structures in the Heisenberg group appear as limit cases. The degeneration of some Thurston’s 3-manifold geometries are studied in this framework. Among other interesting degenerations, the degeneration spherical-Nil-\(\widetilde{SL(2,\mathbb {R})}\) is obtained.