This paper describes cyclic Riemannian Lie groups and their curvatures.
problem Understanding cyclic Riemannian Lie groups and their properties.
method Analyzing left-invariant vector fields and Riemannian metrics.
result Complete description and detailed analysis of cyclic Riemannian Lie groups and their curvatures.
Survey on metrics on compact Lie groups.
problem None explicitly stated; focuses on metrics.
method Left-invariant semi-Riemannian metrics.
result Survey of existing metrics.
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…
Study connects Lie groups to specific Riemannian manifolds.
problem Understanding Lie groups through Riemannian manifold properties.
method Investigates Lie groups as 3D almost paracontact almost paracomplex Riemannian manifolds.
result Established correspondence between Lie algebra and matrix representation.
Using vertical and complete lifts, any left invariant Riemannian metric on a Lie group induces a left invariant Riemannian metric on the tangent Lie group. In the present article we study the Riemannian geometry of tangent bundle of two families of Lie groups. The first one is the family of special Lie groups considere…
Classifies metrics on specific Lie groups.
problem Classifying Riemannian metrics on nonunimodular Lie groups.
method Automorphism classification of inner products on Lie algebras.
result Classification of metrics on 4D nonunimodular Lie groups.
A Lie group G endowed with a left invariant Riemannian metric g is called Riemannian Lie group. Harmonic and biharmonic maps between Riemannian manifolds is an important area of investigation. In this paper, we study different aspects of harmonic and biharmonic homomorphisms between Riemannian Lie groups. We show t…
Study finds specific Lie groups with Kenmotsu structures.
problem Characterizing Lie groups with Kenmotsu structures.
method Determined Lie groups with left invariant Kenmotsu structures.
result These Lie groups are Einstein Riemannian manifolds.
Harmonic maps studied in sub-Riemannian geometry for Lie groups.
problem Defining and characterizing harmonic maps in sub-Riemannian settings.
method Generalization of Riemannian harmonic maps to sub-Riemannian manifolds and Lie groups.
result Conditions for sub-Riemannian harmonic maps and their classification.
New Lie groups generalize H-type groups with nondegenerate centers.
problem Generalizing H-type groups with nondegenerate centers.
method Defined and investigated 2-step nilpotent Lie groups.
result Geometric properties of new Lie groups investigated.
Classifies homogeneous Riemannian structures on 3D Lie groups.
problem Classifying homogeneous Riemannian structures on 3D Lie groups.
method Classification based on left invariant metrics and previous classifications.
result Complete classification of homogeneous Riemannian structures on 3D Lie groups.
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. We investigate contact Lie groups having a left invariant Riemannian or pseudo-Riemannian metric with specific properties such as being bi-invariant, flat, negatively curved, Einstein, etc. We classify some of such contact Lie groups and derive some obstruction results to the existence of left invariant contact structu…
We study Wick-rotations of left-invariant metrics on Lie groups, using results from real GIT (\cite{1}, \cite{2}, \cite{3}). An invariant for Wick-rotation of Lie groups is given, and we describe when a pseudo-Riemannian Lie group can be Wick-rotated to a Riemannian Lie group. We also prove a general version (for gener…
Study finds all isometries for specific Lie groups.
problem Identifying isometry groups in nonunimodular Lie groups.
method Examined left-invariant Riemannian metrics on Lie groups of dimension four.
result Determined full group of isometries for each metric.
Study on curvatures of surfaces in specific Lie groups.
problem Analyzing curvatures of surfaces in 3D contact sub-Riemannian Lie groups.
method Riemannian approximation scheme to derive formulas for curvatures.
result Classification of surfaces with constant horizontal curvatures.
Study on pseudo-Riemannian algebraic Ricci solitons in 4D Lie groups.
problem Investigating conditions for pseudo-Riemannian algebraic Ricci solitons on 4D Lie algebras.
method Analyzing the algebraic Ricci soliton equation for each 4D Lie algebra.
result Complete description of pseudo-Riemannian algebraic Ricci solitons in dimension four.
Compact Lie groups have compact isometry groups with pseudo-Riemannian metrics.
problem Characterizing isometry groups of compact Lie groups with pseudo-Riemannian metrics.
method Analyzing left-invariant pseudo-Riemannian metrics on compact Lie groups.
result Isometry groups of compact Lie groups are compact.
The paper studies Riemannian metrics on Lie groups with specific commutator subgroups.
problem Investigating Riemannian metrics on Lie groups with commutator subgroups of dimensions 1 and 2.
method Explicitly provided Levi-Civita connection, sectional curvature, and Ricci curvature; computed necessary and sufficient conditions for Ricci solitons; characterized Ricci solitons on Lie groups with one-dimensional commutator subgroups; examined indecomposable Lie groups with two-dimensional commutator subgroups.
result Characterization of all Ricci solitons on Lie groups with one-dimensional commutator subgroups and examination of indecomposable Lie groups with two-dimensional commutator subgroups.
New exponential map for Lie groups connects to sub-Riemannian geometry.
problem Developing a new exponential map for Lie groups.
method Introducing a new exponential map related to sub-Riemannian geometry.
result New exponential map connects to sub-Riemannian geometry.
Study of symmetries in 4D Lie groups.
problem Understanding symmetries in specific Lie groups.
method Analyzing isometry groups of left-invariant metrics.
result Full description of isometry groups for 4D Lie groups.
Similarity found in metrics on special Lie groups.
problem Comparing Riemannian metrics on specific Lie groups.
method Proved all metrics are roughly similar via identity.
result All left-invariant Riemannian metrics are roughly similar.
The paper studies geometric properties of tangent Poisson-Lie groups.
problem Geometric properties of tangent Poisson-Lie groups.
method Expressed Levi-Civita connection, curvature, and metacurvature of tangent Poisson-Lie groups in terms of the base group.
result Proved that the space of differential forms on a Poisson-Lie group is a differential graded Poisson algebra if and only if the space on its tangent Poisson-Lie group is.
New metrics with special curvature properties are shown to be parallel in certain Lie groups.
problem Characterizing metrics with harmonic curvature in Lie groups.
method Analyzing left invariant metrics on solvable and low-dimensional Lie groups.
result Left invariant metrics with harmonic curvature are Ricci-parallel in solvable Lie groups and Lie groups of dimension ≤6.
Study classifies biharmonic and harmonic homomorphisms between specific Lie groups.
problem Classifying biharmonic and harmonic homomorphisms between Riemannian three-dimensional unimodular Lie groups.
method Classification based on left invariant Riemannian metrics.
result Classification of biharmonic and harmonic homomorphisms between specific Lie groups.
The paper classifies isometries on specific Lie groups.
problem Classifying isometries on nonnilpotent, solvable 3D Lie groups.
method Proving automorphisms are the only isometries for rank two Almost-Riemannian Structures.
result A classification result for rank two ARSs on nonnilpotent, solvable 3D Lie groups.
Course notes on Lie groups and Riemannian geometry, focusing on applications and low-dimensional examples.
problem Exploring Lie groups and their representations in Riemannian geometry.
method Review of well-known topics and recent advances in Riemannian geometry with symmetries.
result First construction of exceptional holonomy metrics.
Semisimple Lie groups act transitively on pseudo-Riemannian manifolds, making them flat.
problem Understanding actions of semisimple Lie groups on pseudo-Riemannian manifolds.
method Analyzing the pseudo-Riemannian Lichnerowicz conjecture in homogeneous settings.
result Compact pseudo-Riemannian manifolds on which a semisimple group acts conformally, essentially and transitively, are conformally flat.
Study of Riemannian geometry on quaternionic unit ball linked to Sp(1,1) group.
problem Understanding the geometry induced by slice Riemannian metric.
method Developed Lie theoretic study, computed isometry group, compared with quaternionic Poincaré geometry.
result Isometry group of slice Riemannian metric is built from symmetries of Sp(1,1) group.
In this paper we construct infinitely many examples of a Riemannian submersion from a simple, compact Lie group G with bi-invariant metric onto a smooth manifold that cannot be a quotient of G by a group action. This partially addresses a question of K. Grove's about Riemannian submersions from Lie groups.
Study Lie 2-group actions on Riemannian groupoids, proving existence and developing geometric Killing vector fields.
problem Understanding isometric actions of Lie 2-groups on Riemannian groupoids.
method Exhibit properties, prove existence, construct bi-invariant metrics, provide infinitesimal description.
result Existence of 2-equivariant Slice Theorem and Equivariant Tubular Neighborhood Theorem.
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 p-harmonic and harmonic morphisms found on Lie groups.
problem Constructing explicit p-harmonic and harmonic morphisms on Lie groups.
method Using the method of eigenfamilies to construct explicit complex-valued p-harmonic functions and harmonic morphisms.
result Explicit complex-valued p-harmonic functions and harmonic morphisms constructed on non-compact classical Lie groups.
In this paper, we use the powerful tool Milnor bases to classify all the 3−dimensional connected and locally symmetric Riemannian Lie Groups by solving system of polynomial equations of structure constants of each Lie algebra . Moreover, we showed that E0(2), is the only Lie group with locally symmetric left invar…
In these notes we survey basic concepts of affine geometry and their interaction with Riemannian geometry. We give a characterization of affine manifolds which has as counterpart those pseudo-Riemannian manifolds whose Levi-Civita connection is flat. We show that no connected semisimple Lie group admits a left invarian…
Study geodesic orbit property on pseudo-Riemannian H-type nilmanifolds.
problem Characterize geodesic orbit property for pseudo-Riemannian H-type Lie groups.
method Extend results from Riemannian to pseudo-Riemannian H-type Lie groups, focusing on minimal admissible Clifford modules.
result Complete characterization of geodesic orbit property for pseudo-Riemannian H-type Lie groups.
Study left-invariant pseudo-Riemannian metrics on Lie groups focusing on null cone Lie algebras.
problem Characterize left-invariant pseudo-Riemannian metrics on Lie groups in the null cone.
method Use bracket flow on Lie algebra to study metrics on Lie groups.
result Classify all cases of null cone Lie algebras in signatures (1,q) and (2,q).
In this paper we use the Z−decomposition as a tool to find locally symmetric left invariant Riemannian metrics on some Lie groups. For this purpose, we need to compute the spectrum of the curvature operator. Since the study of this spectrum is very difficult, we impose restriction on the class of Riemannian Lie group…
The book explores Lie groups and Carnot-Carathéodory spaces, highlighting their applications in metric geometry and geometric group theory.
problem Exploring non-smooth geometries on Lie groups and their applications.
method Study of left-invariant metrics on Lie groups, focusing on nilpotent and Carnot groups.
result Illustrates the role of metric Lie groups, particularly Carnot groups, in various mathematical contexts.
The paper explores left invariant k-symplectic structures on Lie groups with bi-invariant metrics.
problem Existence and properties of left invariant k-symplectic structures on Lie groups with bi-invariant metrics.
method Analyzing compact semi-simple Lie groups and specific Lie groups with bi-invariant pseudo-Riemannian metrics.
result Compact semi-simple Lie groups and many Lie groups do not carry left invariant k-symplectic structures, except for specific cases.
In the presented paper left-invariant pseudo-Riemannian metrics on four-dimensional Lie groups with zero Schouten-Weyl tensor are investigated. The complete classification of these metric Lie groups is obtained in terms of the structure constants of corresponding Lie algebras.
Study geodesics and shortest arcs on Lie groups with specific metrics.
problem Characterize geodesics and shortest paths on Lie groups with sub-Riemannian metrics.
method Analytical and geometric methods to find geodesics and shortest arcs.
result Found geodesics, shortest arcs, distances, and conjugate loci for specified metrics.
Study Riemann-Finsler geometry on tangent bundles of Lie groups with 2D commutator subgroup.
problem Characterize Riemannian and Finslerian properties of tangent bundles of Lie groups with specific commutator subgroups.
method Investigate sectional curvatures, define Randers metrics, and compute flag curvatures on tangent bundles.
result Explicit formulas for Riemannian curvature tensor on tangent bundles of Lie groups with 2D commutator subgroup.
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.
Sharp estimates on 2-step nilpotent Lie groups' metrics and cones.
problem Estimating asymptotic metrics in 2-step nilpotent Lie groups.
method Developed a novel technique to perturb rectifiable curves.
result Every 2-step nilpotent Riemannian Lie group is at bounded distance from its asymptotic cone.
Study geodesics and shortest arcs on Lie groups with specific metrics.
problem Characterize geodesics and shortest arcs in sub-Riemannian metrics on Lie groups.
method Investigated left-invariant sub-Riemannian metrics on SU(1,1)imesR and SO0(2,1)imesR. result Found geodesics, shortest arcs, cut loci, and conjugate loci.
The paper studies symmetry reduction and optimal control on Riemannian manifolds.
problem Symmetry reduction and optimal control on Riemannian manifolds.
method Derivation of reduced equations of motion for variational problems on Lie groups and Riemannian homogeneous spaces.
result Derivation of geodesic equations and reduced equations of motion for various applications.
Study conformal Killing forms on specific nilpotent Lie groups.
problem Characterize conformal Killing forms on 2-step nilpotent Lie groups.
method Analyzing left-invariant forms on simply connected groups, proving properties of forms based on center dimension.
result Only specific forms exist under certain conditions.