Study proves all left-invariant contact structures on 3D Lie groups are tight.
problem Characterizing tightness of left-invariant contact structures on 3D Lie groups.
method Riemannian methods and unique factorization property for Lie groups.
result All left-invariant contact structures on 3D Lie groups are tight.
This paper studies moduli spaces of statistical structures on Lie groups.
problem Understanding statistical structures on Lie groups.
method Introduced and studied moduli spaces for left-invariant statistical structures on Lie groups.
result Moduli spaces of left-invariant Riemannian metrics are singletons for certain Lie groups.
Study on harmonic spinors on specific Lie groups.
problem Existence of left-invariant harmonic spinors on 3D Lie groups.
method Revised spin Dirac operator formula for left-invariant spinors, identified constraints on Lie algebras, and classified metrics with harmonic spinors.
result Identified conditions and metrics for left-invariant harmonic spinors on 3D Lie groups.
We call a connected Lie group endowed with a left-invariant Lorentzian flat metric Lorentzian flat Lie group. In this Note, we determine all Lorentzian flat Lie groups admitting a timelike left-invariant Killing vector field. We show that these Lie groups are 2-solvable and unimodular and hence geodesically complete. M…
In this paper, we formulate a procedure to obtain a generalization of Milnor frames for left-invariant pseudo-Riemannian metrics on a given Lie group. This procedure is an analogue of the recent studies on left-invariant Riemannian metrics, and is based on the moduli space of left-invariant pseudo-Riemannian metrics. A…
Left-invariant metrics force 2-step nilpotent groups, preserving Kähler-like conditions.
problem Existence of left-invariant pluriclosed Hermitian metrics on Lie groups.
method Analyzing left-invariant metrics on unimodular Lie groups with abelian complex structures.
result Pluriclosed flow preserves Strominger Kähler-like conditions on 2-step nilpotent Lie groups.
Study finds index of symmetry for solvable 3D Lie groups with left-invariant metrics.
problem Determining the index of symmetry for solvable 3D Lie groups with left-invariant metrics.
method Examined all solvable three-dimensional Lie groups, combined with previous work on unimodular groups.
result Index of symmetry is positive for every solvable 3D Lie algebra with a left-invariant metric, and is never 2.
In this paper we consider simply connected Lie groups equipped with left invariant Randers metrics which arise from left invariant Riemannian metrics and left invariant vector fields. Then we study the intersection between automorphism and isometry groups of these spaces. Finally it has shown that for any left invarian…
Classifies left invariant Kundt structures on 3D Lie groups.
problem Understanding Kundt spacetimes and their properties.
method Analyzes local structure and properties of left invariant Kundt structures.
result Classifies all left invariant Kundt structures on 3D simply connected unimodular 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 t…
Study left invariant spray structures on Lie groups, calculating curvature and geodesics.
problem Understanding curvature and geodesics in left invariant spray structures on Lie groups.
method Use invariant frames and canonical bi-invariant Berwald spray structure to analyze left invariant spray structures.
result Established correspondence between geodesics and inverse integral curves of spray vector fields.
Study examines geometry of special metric on Heisenberg group.
problem Geometry of left-invariant Randers metrics on the Heisenberg group.
method Investigation of left-invariant Randers metrics on the Heisenberg group.
result Analysis of the geometry of these metrics.
Study left invariant spray geometry on Lie groups using parallel translations.
problem Understanding parallel translations in left invariant spray geometry.
method Using invariant frames and differential equations on Lie algebra, study parallel translations and curvature.
result Alternative interpretations and proofs of homogeneous curvature formulae.
No left-invariant hypercomplex structures found on compact Lie groups.
problem Existence of left-invariant hypercomplex structures on compact Lie groups.
method Elementary algebraic arguments to show non-existence.
result Compact Lie groups of dimension 4n do not admit left-invariant hypercomplex structures. New method classifies symplectic structures on Lie groups.
problem Classifying left-invariant symplectic structures on Lie groups.
method Using moduli space of left-invariant nondegenerate 2-forms.
result Classified left-invariant symplectic structures on specific Lie groups.
We provide techniques for studying the nonnegatively curved left-invariant metrics on a compact Lie group. For "straight" paths of left-invariant metrics starting at bi-invariant metrics and ending at nonnegatively curved metrics, we deduce a nonnegativity property of the initial derivative of curvature. We apply this …
Study classifies Riemann solitons on specific 3D Lorentzian groups.
problem Classifying Riemann solitons on three-dimensional Lorentzian Lie groups.
method Complete classification through analysis of left-invariant structures.
result Comprehensive classification of Riemann solitons on these groups.
The paper studies Riemannian metrics on tangent Lie groups using two left-invariant metrics.
problem Exploring Riemannian structures on tangent Lie groups.
method Defining a new left-invariant Riemannian metric on the tangent Lie group using two left-invariant metrics and symplectic forms.
result Explicit formulas for the Levi-Civita connection, tensor curvature, and sectional curvature of the new metric in terms of the original metrics.
New method simplifies cohomology computation for specific Lie group structures.
problem Computing cohomology for hypocomplex structures on compact Lie groups.
method Dual of Fréchet-Schwartz spaces theory applied to hypocomplex structures.
result Top-degree cohomology can be computed using only left-invariant forms.
To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi Riemannian metrics on quadratic Lie groups i.e. Lie groups endowed with a bi-invariant semi Riemannia…
The paper classifies left-invariant pseudo-Riemannian metrics on specific Lie groups.
problem Classifying left-invariant pseudo-Riemannian metrics on Lie groups.
method Analyzing left-invariant metrics on specific Lie groups with n≥4. result A complete classification of left-invariant pseudo-Riemannian metrics for Lie groups of dimension n≥4. Study on metrics on specific nilmanifolds, finding new examples and properties.
problem Characteristically solvable nilmanifolds and their metrics.
method Explicit determination of left-invariant metrics and their properties.
result First known examples of Lie groups without positive index of symmetry.
Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
problem Characterize left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
method Analyze left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups, classify Lie algebras and groups.
result New results on left-invariant Lorentzian metrics with harmonic curvature and non-parallel Ricci operator.
New structure found on Lie group tangent bundle.
problem Finding new structures on Lie group tangent bundles.
method Analyzing left invariant structures on Lie groups.
result Tangent bundle of Lie group admits a left-invariant nearly pseudo-Kähler structure.
Counterexample found for Stein property of certain solvable Lie groups.
problem Stein property of simply connected unimodular solvable Lie groups with left-invariant complex structures.
method Constructing a solvable Lie group with specific properties.
result A simply connected solvable Lie group with a left-invariant complex structure whose universal cover is not Stein.
Study on curvatures of specific metric on Heisenberg group.
problem Analyzing curvatures of a specific type of metric on a Heisenberg group.
method Examined left invariant Z-Randers metrics on the five-dimensional Heisenberg group.
result Existence of flags with strictly negative and strictly positive curvatures.
We determine all Ricci flat left invariant Lorentzian metrics on simply connected 2-step nilpotent Lie groups. We show that the 2k+1-dimensional Heisenberg Lie group H2k+1 carries a Ricci flat left invariant Lorentzian metric if and only if k=1. We show also that for any 2≤q≤k, H2k+1 carries a R…
Research explores Kähler and semi-para-Kähler structures on specific Lie groups.
problem Existence of Kähler and semi-para-Kähler structures on six-dimensional unsolvable Lie groups.
method Examines four specific Lie algebras and their structures.
result One Lie algebra admits Kähler metrics, others admit semi-para-Kähler and semi-Kähler structures.
Study odd generalized Einstein metrics on 3D Lie groups.
problem Classify odd generalized Einstein metrics on 3D Lie groups.
method Left-invariant generalized connections, divergence operators, and Ricci tensors.
result Describe all odd generalized Einstein metrics on all 3D Lie groups.
Study of special geometric structures on Lie groups.
problem Investigating left-invariant mG2∗-structures with specific holonomy properties. method Classification of indecomposable holonomy algebras, determination of infinitesimal holonomy algebras.
result Only abelian subalgebras of dimension 2 or 3 arise as holonomy algebras.
We prove that any real Lie group of dimension \leq 5 admits a left invariant flat projective structure. We also prove that a real Lie group L of dimension \leq 5 admits a left invariant flat affine structure if and only if the Lie algebra of L is not perfect.
Study finds optimal loops in hyperbolic space with Finsler structure.
problem Optimal loops in Finsler hyperbolic plane.
method Left-invariant Finsler structure, convex trigonometry functions.
result Optimal isoperimetric loops found in terms of trigonometry functions.
All known examples of homogeneous Einstein metrics of negative Ricci curvature can be realized as left-invariant Riemannian metrics on solvable Lie groups. After defining a notion of maximal symmetry among left-invariant Riemannian metrics on a Lie group, we prove that any left-invariant Einstein metric of negative Ric…
Study magnetic curvature on Lie groups, extending Milnor's work.
problem Exploring magnetic curvatures on Lie groups.
method Computing magnetic curvatures and analyzing algebraic properties.
result Extending results from Milnor's classic paper on left-invariant metrics.
This study broadens understanding of metric groups on Lie groups.
problem Understanding the relationship between automorphism and isometry groups of (α,β)-metrics. method Generalizing from Randers metrics to (α,β)-metrics, analyzing automorphism and isometry groups of Finsler manifolds. result For any left invariant vector field and compact subgroup of automorphisms, there exists an (α,β)-metric with the subgroup as a subgroup of its isometry group. With a f-left-invariant Riemannian metric on a Lie group G, we mean a Riemannian metric which is conformally equivalent to a left-invariant Riemannian metric, with the conformal factor f. In this article, we study the geometry of such metrics and give a necessary and sufficient condition for an f-left-invariant Rie…
Study on special tensors in specific geometric spaces.
problem Characterizing symmetric Killing tensors on nilmanifolds.
method Investigated left-invariant symmetric Killing 2-tensors on 2-step nilpotent Lie groups with a Riemannian metric.
result Found genuine examples of symmetric Killing tensors that are not simple combinations.
Study on 3D Lie groups' symmetry index, proving positive index for all.
problem Determining the index of symmetry in 3D Lie groups.
method Analyzing left-invariant metrics and quotient geometry.
result Positive index of symmetry for all 3D unimodular Lie groups.
We discuss the geometry of homogeneous Ricci solitons. After showing the nonexistence of compact homogeneous and noncompact 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 l…
The goal of this paper is to provide a method, based on the theory of extensions of left-symmetric algebras, for classifying left-invariant affine structures on a given solvable Lie group of low dimension. To better illustrate our method, we shall apply it to classify complete left-invariant affine structures on the os…
Computational techniques calculate dimensions of complex structures.
problem Calculating dimensions of complex structures on manifolds.
method Developed computational techniques to calculate Kodaira dimension and Dolbeault harmonic forms.
result Computed dimensions of left-invariant almost complex structures.
In the previous paper [MR2430243] we computed some geometric quantities such as curvature and flag curvature for a general left invariant Finsler metric on a two-step nilpotent group. In the present paper we give a more complete description of the Chern--Rund connection defined by a left invariant Randers metric on the…
Generalizes Dolbeault cohomology computation to Levi-flat CR structures on compact Lie groups.
problem Computing Dolbeault cohomology for Levi-flat CR structures on compact Lie groups.
method Algebraic classification of left-invariant CR structures combined with Pittie's result on compact Lie groups.
result Generalization of Dolbeault cohomology computation to Levi-flat CR structures.
Study of Riemann-Poisson Lie groups with compatibility conditions.
problem Characterizing and constructing Riemann-Poisson Lie groups.
method Left-invariant metrics and Poisson tensors compatible in Lie groups.
result Characterization and construction of Lie algebras up to dimension 5.
Classifies Killing forms of arbitrary degree on specific nilpotent Lie groups.
problem Classifying Killing forms of arbitrary degree on specific Lie groups.
method Analyzing left-invariant Killing forms on simply connected 2-step nilpotent Lie groups with left-invariant metrics.
result Classified Killing forms when center is at most 2-dimensional.
This work reviews left-invariant optimal control problems on Lie groups.
problem Optimal control problems on Lie groups with big symmetry.
method Review of main notions, methods, and results.
result Description of extremal trajectories and their optimality, cut time and cut locus, optimal synthesis.
A n-dimensional Lie group G equipped with a left invariant symplectic form $\om^+$ is called a symplectic Lie group. It is well-known that $\om^+$ induces a left invariant affine structure on G. Relatively to this affine structure we show that the left invariant Poisson tensor π+ corresponding to $\om^+$ is po…
Researchers find optimal paths on a specific geometric group.
problem Finding optimal paths on a Cartan group with a sub-Finsler quasimetric.
method Using the Pontryagin Maximum Principle in coordinates of the first kind.
result They found extremals for arbitrary left-invariant sub-Finsler quasimetrics.