Non-unimodular Lie 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.

Three-Dimensional Nonunimodular Lie Groups with a Riemannian Metric of an Invariant Ricci Soliton and a Semisymmetric Metric Connection

Three-Dimensional Nonunimodular Lie Groups with a Riemannian Metric of an Invariant Ricci Soliton and a Semisymmetric Metric Connection

Isometry groups of three-dimensional Lie groups

We compute the full isometry group of any left-invariant metric on a simply connected, non-unimodular Lie group of dimension three. As an application, we determine the index of symmetry of such metrics and prove that the singularities of the moduli space of left-invariant metrics, up to isometric automorphism are contained in the subspace of classes of metrics with maximal index of symmetry.

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.

∗-Ricci tensor on almost Kenmotsu 3-manifolds

In this paper, we obtain the expressions of the ∗-Ricci operator of a three-dimensional almost Kenmotsu manifold [Formula: see text] and find that the ∗-Ricci tensor is not symmetric for [Formula: see text]. We obtain a necessary and sufficient condition for the ∗-Ricci tensor to be symmetric and proved that if the ∗-Ricci tensor of a non-Kenmotsu almost Kenmotsu 3-[Formula: see text]-manifold [Formula: see text] is symmetric, then [Formula: see text] is locally isometric to a three-dimensional non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure. Further, it is shown that the ∗-Ricci tensor of a non-Kenmotsu almost Kenmotsu 3-manifold [Formula: see text] is parallel if and only if [Formula: see text] is ∗-Ricci flat. In addition, [Formula: see text] satisfying [Formula: see text] is locally isometric to [Formula: see text]. Finally, we discuss about [Formula: see text]-parallel ∗-Ricci tensor on almost Kenmotsu 3-manifolds.

Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators

Abstract In this paper, it is proved that the Ricci operator of an almost Kenmotsu 3-h-manifold M is of transversely Killing-type if and only if M is locally isometric to the hyperbolic 3-space ℍ 3 ( − 1 ) {{\mathbb{H}}}^{3}(-1) or a non-unimodular Lie group endowed with a left invariant non-Kenmotsu almost Kenmotsu structure. This result extends those results obtained by Cho [Local symmetry on almost Kenmotsu three-manifolds, Hokkaido Math. J. 45 (2016), no. 3, 435–442] and Wang [Three-dimensional locally symmetric almost Kenmotsu manifolds, Ann. Polon. Math. 116 (2016), no. 1, 79–86; Three-dimensional almost Kenmotsu manifolds with η \eta -parallel Ricci tensor, J. Korean Math. Soc. 54 (2017), no. 3, 793–805].

Cyclic-parallel Ricci tensors on a class of almost Kenmotsu 3-manifolds

In this paper, we give a local characterization for the Ricci tensor of an almost Kenmotsu [Formula: see text]-manifold [Formula: see text] to be cyclic-parallel. As an application, we prove that if [Formula: see text] has cyclic-parallel Ricci tensor and satisfies [Formula: see text], (where [Formula: see text] is the Lie derivative of [Formula: see text] along the Reeb flow and both [Formula: see text] and [Formula: see text] are smooth functions such that [Formula: see text] is invariant along the contact distribution), then [Formula: see text] is locally isometric to either the hyperbolic space [Formula: see text] or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure.

Grassmann geometry on the 3-dimensional non-unimodular Lie groups

We study the Grassmann geometry of surfaces when the ambient space is a 3-dimensional non-unimodular Lie group with left invariant metric. This work together with our previous papers yield a complete classification of Grassmann geometry of orbit type in all 3-dimensional homogeneous spaces.

Pseudo-differential Operators, Wigner Transforms and Weyl Transforms on the Poincaré Unit Disk

Using the affine group and the Cayley transform from the unit disk $${{\mathbb {D}}}$$ onto the upper half plane, we can turn $${{\mathbb {D}}}$$ into a group, which we call the Poincare unit disk. With this construction, $${{\mathbb {D}}}$$ is a noncompact and nonunimodular Lie group. We characterize all infinite-dimensional, irreducible and unitary representations of $${{\mathbb {D}}}$$ . By means of these representations, the Fourier transform on $${{\mathbb {D}}}$$ is defined. The Plancherel theorem and hence the Fourier inversion formula can be given. Then pseudo-differential operators with operator-valued symbols, operator-valued Wigner transforms, and Weyl transforms on $${{\mathbb {D}}}$$ are defined.

Ricci solitons on almost Kenmotsu 3-manifolds

Abstract Let (M3, g) be an almost Kenmotsu 3-manifold such that the Reeb vector field is an eigenvector field of the Ricci operator. In this paper, we prove that if g represents a Ricci soliton whose potential vector field is orthogonal to the Reeb vector field, then M3 is locally isometric to either the hyperbolic space ℍ3(−1) or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure. In particular, when g represents a gradient Ricci soliton whose potential vector field is orthogonal to the Reeb vector field, then M3 is locally isometric to either ℍ3(−1) or ℍ2(−4) × ℝ.

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.

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

Solvable Lie algebras are not that hypo

We study a type of left-invariant structure on Lie groups or, equivalently, on Lie algebras. We introduce obstructions to the existence of a hypo structure, namely the five-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3). The choice of a splitting $ {\mathfrak{g}^*} = {V_1} \oplus {V_2} $ , and the vanishing of certain associated cohomology groups, determine a first obstruction. We also construct necessary conditions for the existence of a hypo structure with a fixed almost-contact form. For nonunimodular Lie algebras, we derive an obstruction to the existence of a hypo structure, with no choice involved. We apply these methods to classify solvable Lie algebras that admit a hypo structure.

Vacuum polarization of a scalar field on the nonunimodular lie groups

A method for calculating the vacuum average energy-momentum tensors for a scalar field on the nonunimodular Lie groups is developed. To solve this problem, a method of generalized harmonic analysis on the Lie groups is used based on the method of orbits of coadjoint representation.

Left-invariant Riemannian metrics on four-dimensional nonunimodular Lie groups with zero-divergence Weyl tensor

Left-invariant Riemannian metrics on four-dimensional nonunimodular Lie groups with zero-divergence Weyl tensor

The signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional lie groups. The unimodular case

We obtain a complete classification of all the possible values of the signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional nonunimodular Lie groups.

Homogeneous geodesics of non-unimodular Lorentzian Lie groups and naturally reductive Lorentzian spaces in dimension three

AbstractWe determine, for all three-dimensional non-unimodular Lie groups equipped with a Lorentzian metric, the set of homogeneous geodesics through a point. Together with the results of [G. Calvaruso, Homogeneous structures on three-dimensional Lorentzian manifolds.J. Geom. Phys.57(2007), 1279–1291. MR2287304 (2008b:53066) Zbl 1112.53051] and [G. Calvaruso, R. A. Marinosci, Homogeneous geodesics of three-dimensional unimodular Lorentzian Lie groups.Mediterr. J. Math.3(2006), 467–481. MR2274738 Zbl pre05151042], this leads to the full classification of three-dimensional Lorentzian g.o. spaces and naturally reductive spaces.

Polar Wavelet Transforms and Localization Operators

The notion of a polar wavelet transform is introduced. The underlying non-unimodular Lie group, the associated square-integrable representations and admissible wavelets are studied. The resolution of the identity formula for the polar wavelet transform is then formulated and proved. Localization operators corresponding to the polar wavelet transforms are then defined. It is proved that under suitable conditions on the symbols, the localization operators are, in descending order of complexity, paracommutators, paraproducts and Fourier multipliers.

Method of orbits of coassociated representation in thermodynamics of the lie noncompact groups

In the present paper, an efficient method of solving the main problem of thermodynamics of homogeneous spaces is presented. The method is based on the formalism of the noncommutative harmonic analysis relying on the method of orbits of coassociated representation. In the present work, the formula is derived that allows one to construct effectively the density matrix and the statistical sum in space of any arbitrary generally noncompact non-unimodular Lie group. To illustrate the method, an example is given of exact calculation of the statistical sum and density matrix in the Riemannian space of the noncompact Lie group with left-invariant metric.

Non-unimodular Lie foliations

Let F be a G-Lie foliation on a compact manifold M. If F is not unimodular then either M or the closures of the leaves fiber over S 1 . To cite this article: E. Macias-Virgós, P. Martín-Méndez, C. R. Acad. Sci. Paris, Ser. I 340 (2005).