Left-invariant Metrics Research Articles

Rough similarity of left-invariant Riemannian metrics on some Lie groups

We consider Lie groups that are either Heintze groups or Sol-type groups, which generalize the three-dimensional Lie group SOL. We prove that all left-invariant Riemannian metrics on each such a Lie group are roughly similar via the identity map. This allows us to reformulate in a common framework former results by Le Donne–Xie, Eskin–Fisher–Whyte, Carrasco Piaggio, and recent results of Ferragut and Kleiner–Müller–Xie, on quasi-isometries of these solvable groups.

A harmonic property of right invariant priors

AbstractIt is shown that, on any Lie group, the density ratio of the right invariant measure to the left invariant measure is harmonic with respect to the left invariant Riemannian metric. This result is applied to the Bayesian prediction theory on group invariant statistical models. A method of constructing Bayesian prior distributions that asymptotically dominate the right invariant priors is provided.

The Heterotic-Ricci Flow and Its Three-Dimensional Solitons

We introduce a novel curvature flow, the Heterotic-Ricci flow, as the two-loop renormalization group flow of the Heterotic string common sector and study its three-dimensional compact solitons. The Heterotic-Ricci flow is a coupled curvature evolution flow, depending on a non-negative real parameter κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}, for a complete Riemannian metric and a three-form H on a manifold M. Its most salient feature is that it involves several terms quadratic in the curvature tensor of a metric connection with skew-symmetric torsion H. When κ=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa = 0$$\\end{document} the Heterotic-Ricci flow reduces to the generalized Ricci flow and hence it can be understood as a modification of the latter via the second-order correction prescribed by Heterotic string theory, whereas when H=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H=0$$\\end{document} and κ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa >0$$\\end{document} the Heterotic-Ricci flow reduces to a constrained version of the RG-2 flow and hence it can be understood as a generalization of the latter via the introduction of the three-form H. Solutions of Heterotic supergravity with trivial gauge bundle, which we call Heterotic solitons, define a particular class of three-dimensional solitons for the Heterotic-Ricci flow and constitute our main object of study. We prove a number of structural results for three-dimensional Heterotic solitons, obtaining, in particular, the complete classification of compact three-dimensional strong Heterotic solitons as hyperbolic three-manifolds or quotients of the Heisenberg group equipped with a left-invariant metric. Furthermore, we prove that all Einstein three-dimensional Heterotic solitons have constant dilaton and leave as open the construction of a Heterotic soliton with non-constant dilaton. In this direction, we prove that Einstein Heterotic solitons with constant dilaton are rigid and therefore cannot be deformed into a solution with non-constant dilaton. This is, to the best of our knowledge, the first rigidity result for compact supergravity solutions in the literature.

Conformal vector fields on Lie groups: The trans-Lorentzian signature

A pseudo-Riemannian Lie group is a connected Lie group endowed with a left-invariant pseudo-Riemannian metric of signature (p,q). In this paper, we study pseudo-Riemannian Lie groups (G,〈⋅,⋅〉) with non-Killing left-invariant conformal vector fields. Firstly, we prove that if h is a Cartan subalgebra for a semisimple Levi factor of the Lie algebra g, then dim⁡h≤max⁡{0,min⁡{p,q}−2}. Secondly, we classify trans-Lorentzian Lie groups (i.e., min⁡{p,q}=2) with non-Killing left-invariant conformal vector fields, and prove that [g,g] is at most three-step nilpotent. Thirdly, based on the classification of the trans-Lorentzian Lie groups, we show that the corresponding Ricci operators are nilpotent, and consequently the scalar curvatures vanish. As a byproduct, we prove that four-dimensional trans-Lorentzian Lie groups with non-Killing left-invariant conformal vector fields are necessarily conformally flat, and construct a family of five-dimensional ones which are not conformally flat.

A HIERARCHY ON NON-ARCHIMEDEAN POLISH GROUPS ADMITTING A COMPATIBLE COMPLETE LEFT-INVARIANT METRIC

Abstract In this article, we introduce a hierarchy on the class of non-archimedean Polish groups that admit a compatible complete left-invariant metric. We denote this hierarchy by $\alpha $ -CLI and L- $\alpha $ -CLI where $\alpha $ is a countable ordinal. We establish three results: (1) G is $0$ -CLI iff $G=\{1_G\}$ ; (2) G is $1$ -CLI iff G admits a compatible complete two-sided invariant metric; and (3) G is L- $\alpha $ -CLI iff G is locally $\alpha $ -CLI, i.e., G contains an open subgroup that is $\alpha $ -CLI. Subsequently, we show this hierarchy is proper by constructing non-archimedean CLI Polish groups $G_\alpha $ and $H_\alpha $ for $\alpha <\omega _1$ , such that: (1) $H_\alpha $ is $\alpha $ -CLI but not L- $\beta $ -CLI for $\beta <\alpha $ ; and (2) $G_\alpha $ is $(\alpha +1)$ -CLI but not L- $\alpha $ -CLI.

Spontaneously interacting qubits from Gauss-Bonnet

Building on previous constructions examining how a collection of small, locally interacting quantum systems might emerge via spontaneous symmetry breaking from a single-particle system of high dimension, we consider a larger family of geometric loss functionals and explicitly construct several classes of critical metrics which “know about qubits” (KAQ). The loss functional consists of the Ricci scalar with the addition of the Gauss-Bonnet term, which introduces an order parameter that allows for spontaneous symmetry breaking. The appeal of this method is two-fold: (i) the Ricci scalar has already been shown to have KAQ critical metrics and (ii) exact equations of motions are known for loss functionals with generic curvature terms up to two derivatives. We show that KAQ critical metrics, which are solutions to the equations of motion in the space of left-invariant metrics with fixed determinant, exist for loss functionals that include the Gauss-Bonnet term. We find that exploiting the subalgebra structure leads us to natural classes of KAQ metrics which contain the familiar distributions (GUE, GOE, GSE) for random Hamiltonians. We introduce tools for this analysis that will allow for straightfoward, although numerically intensive, extension to other loss functionals and higher-dimension systems.

Reduction of Sufficient Conditions in Variational Obstacle Avoidance Problems

This paper studies Sufficient conditions in a variational obstacle avoidance problem on complete Riemannian manifolds. That is, we minimize an action functional, among a set of admissible curves, which depends on an artificial potential function used to avoid obstacles. We provide necessary and Sufficient conditions under which the resulting critical points—the so-called modified Riemannian cubics—are local minimizers. We then study the theory of reduction by symmetries of Sufficient conditions for optimality in variational obstacle avoidance problems on Lie groups endowed with a left-invariant metric. This amounts to left-translating the Bi-Jacobi fields described to the Lie algebra, and studying the corresponding bi-conjugate points. New conditions are provided in terms of the invertibility of a certain matrix.

Полная группа изометрий специальной трёхмерной группы Ли

We consider a special three-dimensional Lie algebra 3 of Bianchi type V. A matrix representation of this Lie algebra and the corresponding connected simply connected Lie group 𝑆3 is found and natural coordinates are introduced, which are determined by the matrix representation. Formulas for exponential mapping with respect to natural coordinates are found. Complete groups of autoisometries of the special Lie algebra with respect to the Euclidean or Lorentz scalar product are written out. The leftinvariant Riemannian metric of the Lie group 𝑆3 and the complete group of isometries of the resulting homogeneous manifold are found. This manifold turned out to be a space of constant Ricci curvature.

CYT and SKT Metrics on Compact Semi-Simple Lie Groups

A Hermitian metric on a complex manifold $(M, I)$ of complex dimension $n$ is called Calabi-Yau with torsion (CYT) or Bismut-Ricci flat, if the restricted holonomy of the associated Bismut connection is contained in ${\rm SU}(n)$ and it is called strong K\"ahler with torsion (SKT) or pluriclosed if the associated fundamental form $F$ is $\partial \overline \partial$-closed. In the paper we study the existence of left-invariant SKT and CYT metrics on compact semi-simple Lie groups endowed with a Samelson complex structure $I$. In particular, we show that if $I$ is determined by some maximal torus $T$ and $g$ is a left-invariant Hermitian metric, which is also invariant under the right action of the torus $T$, and is both CYT and SKT, then $g$ has to be Bismut flat.

Left invariant Ricci flat metrics on Lie groups

Abstract In this paper, we apply the double extension process to study left invariant Ricci flat metrics on solvable and non-solvable Lie groups. An inductive method to produce new Ricci flat metrics from the old ones is established. As applications, we prove the following two results: (i) Every nilpotent Lie group with dim ⁡ C ⁢ ( G ) ≥ 1 2 ⁢ ( dim ⁡ G - 1 ) {\dim\mathrm{C}(G)\geq\frac{1}{2}(\dim G-1)} admits a left invariant Ricci flat metric. (ii) Given a Lie group G, there exists a nilpotent Lie group N with nilpotent index at most 2 such that G × N {G\times N} admits a left invariant Ricci flat metric. We also construct infinitely many new explicit examples of left invariant Ricci flat metrics on nilpotent Lie groups.

On the geometry of the Heisenberg group with a balanced metric

We study the geometry of the Heisenberg group Nil3 with a balanced metric, the sum of the left and right invariant metrics. We prove that with this metric, Nil3 splits as a Riemannian product T×Z, where T is a totally geodesic surface and Z the center of Nil3. So we prove the existence of complete properly embedded minimal surfaces in Nil3 by solving the asymptotic Dirichlet problem for the minimal surface equation on T. We also show the existence of complete properly embedded minimal surfaces foliating an open set of Nil3 having as boundary a given curve Γ in T, satisfying the exterior circle condition, by solving the exterior Dirichlet problem for the minimal surface equation in the unbounded connected component of T∖Γ.

Pseudo-Kähler and pseudo-Sasaki structures on Einstein solvmanifolds

The aim of this paper is to construct left-invariant Einstein pseudo-Riemannian Sasaki metrics on solvable Lie groups. We consider the class of mathfrak {z}-standard Sasaki solvable Lie algebras of dimension 2n+3, which are in one-to-one correspondence with pseudo-Kähler nilpotent Lie algebras of dimension 2n endowed with a compatible derivation, in a suitable sense. We characterize the pseudo-Kähler structures and derivations giving rise to Sasaki–Einstein metrics. We classify mathfrak {z}-standard Sasaki solvable Lie algebras of dimension le 7 and those whose pseudo-Kähler reduction is an abelian Lie algebra. The Einstein metrics we obtain are standard, but not of pseudo-Iwasawa type.

Eigenvalues of Ricci Operator of Four-Dimensional Locally Homogeneous Riemannian Manifolds with Nontrivial Isotropy Subgroup

The topology of Riemannian manifolds can be linked to the eigenvalues of curvature operators, which was demonstrated in the works of J. Milnor, V.N. Berestovsky, V.V. Slavkii, E.D. Rodionov, and Yu.G. Nikonorov. J. Milnor studied the eigenvalues of the Ricci curvature operator of left-invariant Riemannian metrics on Lie groups, and identified possible signatures of the Ricci operator for three-dimensional Lie groups. O. Kowalski and S. Nikcevic later resolved the problem of prescribed spectrum values of the Ricci operator on three-dimensional metric Lie groups and Riemannian locally homogeneous spaces. D.N. Oskorbin, E.D. Rodionov, and O.P. Khomova also obtained similar results for the one-dimensional curvature operator and the sectional curvature operator. A.G. Kremlev and Yu.G. Nikonorov investigated the fourdimensional case and studied the possible signatures of the Ricci curvature of left-invariant Riemannian metrics on Lie groups. In this study, we aim to solve the problem of prescribed eigenvalues of the Ricci operator on locally homogeneous Riemannian manifolds with a nontrivial isotropy subgroup.

Mollifications of contact mappings of Engel group

The contact mappings belonging to the metric Sobolev classes are studied on an Engel group with a left-invariant sub-Riemannian metric. In the Euclidean space one of the main methods to handle non-smooth mappings is the mollification, i.e., the convolution with a smooth kernel. An extra difficulty arising with contact mappings of Carnot groups is that the mollification of a contact mapping is usually not contact. Nevertheless, in the case considered it is possible to estimate the magnitude of deviation of contactness sufficiently to obtain useful results. We obtain estimates on convergence (or sometimes divergence) of the components of the differential of the mollified mapping to the corresponding components of the Pansu differential of the contact mapping. As an application to the quasiconformal analysis, we present alternative proofs of the convergence of mollified horizontal exterior forms and the commutativity of the pull-back of the exterior form by the Pansu differential with the exterior differential in the weak sense. These results in turn allow us to obtain such basic properties of mappings with bounded distortion as H\"older continuity, differentiability almost everywhere in the sense of Pansu, Luzin $\mathcal{N}$-property.

Geometry of cotangent bundle of Heisenberg group

In this paper, the classification of left invariant Riemannian metrics on the cotangent bundle of the (2n+1)-dimensional Heisenberg group up to the action of the automorphism group is presented. Moreover, it is proved that the complex structure on this group is unique, and the corresponding pseudo-Kähler metrics are described and shown to be Ricci flat. It is known that this algebra admits an ad-invariant metric of neutral signature. Here, the uniqueness of such metric is proved.

Geodesic completeness of pseudo and holomorphic-Riemannian metrics on Lie groups

This paper is devoted to geodesic completeness of left-invariant metrics for real and complex Lie groups. We start by establishing the Euler–Arnold formalism in the holomorphic setting. We study the real Lie group SL(2,R) and reobtain the known characterization of geodesic completeness and, in addition, present a detailed study where we investigate the maximum domain of definition of every single geodesic for every possible metric. We investigate completeness and semicompleteness of the complex geodesic flow for left-invariant holomorphic metrics and, in particular, establish a full classification for the Lie group SL(2,ℂ).

A classification of left-invariant Lorentzian metrics on some nilpotent Lie groups

It has been known that there exist exactly three left-invariant Lorentzian metrics up to scaling and automorphisms on the three dimensional Heisenberg group. In this paper, we classify left-invariant Lorentzian metrics on the direct product of three dimensional Heisenberg group and the Euclidean space of dimension $n-3$ with $n \geq 4$, and prove that there exist exactly six such metrics on this Lie group up to scaling and automorphisms. Moreover we show that only one of them is flat, and the other five metrics are Ricci solitons but not Einstein. We also characterize this flat metric as the unique closed orbit, where the equivalence class of each left-invariant metric can be identified with an orbit of a certain group action on some symmetric space.

Critical metrics for quadratic curvature functionals on some solvmanifolds

We prove the existence of four-dimensional compact manifolds admitting some non-Einstein Lorentzian metrics, which are critical points for all quadratic curvature functionals. For this purpose, we consider left-invariant semi-direct extensions G_{mathcal S}=H rtimes exp ({mathbb {R}}S) of the Heisenberg Lie group H, for any mathcal S in {mathfrak {s}}{mathfrak {p}}(1,mathbb R), equipped with a family g_a of left-invariant metrics. After showing the existence of lattices in all these four-dimensional solvable Lie groups, we completely determine when g_a is a critical point for some quadratic curvature functionals. In particular, some four-dimensional solvmanifolds raising from these solvable Lie groups admit non-Einstein Lorentzian metrics, which are critical for all quadratic curvature functionals.

Geometry of Tangent Poisson–Lie Groups

Let G be a Poisson–Lie group equipped with a left invariant contravariant pseudo-Riemannian metric. There are many ways to lift the Poisson structure on G to the tangent bundle TG of G. In this paper, we induce a left invariant contravariant pseudo-Riemannian metric on the tangent bundle TG, and we express in different cases the contravariant Levi-Civita connection and curvature of TG in terms of the contravariant Levi-Civita connection and the curvature of G. We prove that the space of differential forms Ω*(G) on G is a differential graded Poisson algebra if, and only if, Ω*(TG) is a differential graded Poisson algebra. Moreover, we show that G is a pseudo-Riemannian Poisson–Lie group if, and only if, the Sanchez de Alvarez tangent Poisson–Lie group TG is also a pseudo-Riemannian Poisson–Lie group. Finally, some examples of pseudo-Riemannian tangent Poisson–Lie groups are given.

Almost Kähler structures on complex hyperbolic space

Complex hyperbolic space has the structure of a solvable Lie group. Based on the recent classification of Riemannian left-invariant metrics we describe all left-invariant almost K?hler structures on that group. The only K?hler structure is the standard structure of the complex hyperbolic space and all others are strictly almost K?hler. We calculated the Chern connection and obtained the characterization by its torsion tensor. It is proved that the only SCF-soliton is the K?hler one.