Study Einstein Lie groups and geodesic orbit manifolds, finding some are not geodesic orbit.
problem Characterizing Einstein Lie groups and geodesic orbit manifolds.
method Characterizing GimesK-invariant geodesic orbit metrics on Lie groups G for regular subgroups K. result Extensive classes of compact simple Einstein Lie groups are not geodesic orbit manifolds.
Found a 6D Lie group with a closed geodesic.
problem Existence of closed geodesics on homogeneous Riemannian manifolds.
method Provided a specific Lie group example.
result Answered affirmatively to a question about closed geodesics on homogeneous spaces.
The study constructs families of nilpotent Lie groups with geodesic orbit metrics.
problem Finding geodesic orbit metrics on nilpotent Lie groups.
method Construction of continuous families of nilpotent Lie groups.
result Continuous families of non-isomorphic nilpotent Lie groups with geodesic orbit metrics.
This paper classifies geodesic orbit metrics on compact Lie group G2.
problem Classifying geodesic orbit metrics on compact Lie groups.
method Using representation theory of Lie subgroups, specifically weakly regular subgroups.
result Left-invariant geodesic orbit metrics on compact Lie group G2 are classified. Study geodesic Lie groups' convergence to limits with quantitative estimates.
problem Quantifying convergence rates of geodesic Lie groups to their limits.
method Estimates on the difference between original metrics and asymptotic/tangent metrics.
result Sharpens existing bounds on convergence rates.
The paper studies geodesic completeness for Lie groups and their metrics.
problem Geodesic completeness of pseudo and holomorphic Riemannian metrics on Lie groups.
method Euler-Arnold formalism, detailed study of geodesics, classification of metrics.
result Full classification of geodesic completeness for the Lie group SL(2, C).
Solves geodesic completeness on pseudo-homothetic Lie group.
problem Geodesic completeness on pseudo-homothetic Lie group.
method Exhibited a family of complete metrics with bounded geodesic velocity.
result Set of complete metrics is not closed.
Study of Gödel Universe as Lie group with specific metric.
problem Characterize geodesics in the Gödel Universe.
method Geometric theory of optimal control applied to Lie groups with left-invariant Lorentz metrics.
result No closed timelike or isotropic geodesics in the Gödel Universe.
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.
Researchers classify geodesic orbit spaces for compact Lie groups of rank two.
problem Identifying geodesic orbit spaces for compact Lie groups of specific rank.
method Classification of simply connected geodesic orbit spaces where G is a compact Lie group of rank two.
result Only certain spheres and projective spaces, with metrics induced from Hopf fibrations, are geodesic orbit spaces for compact Lie groups of rank two.
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.
In this paper, we investigate left-invariant geodesic orbit metrics on connected simple Lie groups, where the metrics are formed by the structures of generalized flag manifolds. We prove that all these left-invariant geodesic orbit metrics on simple Lie groups are naturally reductive.
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.
The paper examines geodesic completeness in Lie groups with specific vector fields.
problem Investigating geodesic completeness in Lie groups with special vector fields.
method Analyzing left-invariant Lorentzian metrics on simple Lie groups with Killing vector fields.
result Conditions for geodesic completeness in Lie groups with specific vector fields.
Average signature measures geodesics in Lie groups.
problem Understanding geometric properties of Lie groups through geodesic paths.
method Introducing average signature A(G) and using it with trace operation to recover geometric properties. result Average signature can recover geometric properties like dimension, diameter, volume, and scalar curvature.
We show that strictly abnormal geodesics arise in graded nilpotent Lie groups. We construct such a group, for which some Carnot geodesics are strictly abnormal; in fact, they are not normal in any subgroup. In the step-2 case we also prove that these geodesics are always smooth. Our main technique is based on the equat…
Periodic geodesics on Hilbert half-Lie groups exist whenever the fundamental group is nontrivial.
problem Existence of periodic geodesics on Hilbert half-Lie groups
method Using completeness results and Lyusternik-Fet type theorem
result Periodic geodesics exist whenever the fundamental group is nontrivial
In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is possible to id…
We give a complete list of those left invariant unit vector fields on three-dimensional Lie groups with the left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group with the Sasaki metric. As a result, each class of three-dimensional Lie groups admits the totally geodesic…
We extend Agrachev-Brockett-Jurjdevic's solution to normal sub-Riemannian geodesics.
problem Finding normal sub-Riemannian geodesics in Lie group structures.
method Constructing sub-Riemannian structures from Lie subalgebra filtrations and applying to homogeneous spaces.
result Explicit solutions for normal geodesics in general chains of Lie subgroups.
On the ground of origins of the theory of Lie groups and Lie algebras, their (co)adjoint representations, and the Pontryagin maximum principle for the time-optimal problem are given an independent foundation for methods of geodesic vector field to search for normal geodesics of left-invariant (sub-)Finsler metrics on L…
Study isotropic embeddings of Lie group orbits and their application to magnetic geodesic flows.
problem Embedding coadjoint orbits and their equivalence to magnetic geodesic flows.
method Review and initiate study of isotropic and Lagrangian embeddings for SO and Sp cases, then apply to magnetic geodesic flows. result Equivalence between magnetic geodesic flows and certain spin chains.
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.
In this paper we describe the geodesics of a left-invariant sub-Riemannian metric on the three-dimensional solvable Lie group SOLV−.
Minimal number of geodesics in Finsler manifolds with indefinite Killing form is at least four.
problem Determining the minimal number of homogeneous geodesics in Finsler manifolds with indefinite Killing form.
method Analyzing examples of Lie groups with invariant Finsler metrics and presenting new examples.
result Homogeneous Finsler manifolds with indefinite Killing form admit at least four homogeneous geodesics.
We study the geodesics problem in Heisenberg group H (case SR and riemannian). The sheaf of infinitesimal automorphisms of the (2n,2n+1) distribution D over H is an infinite, transitive Lie algebra sheaf.
For a Riemannian submersion from a simple compact Lie group with a bi-invariant metric, we prove the action of its holonomy group on the fibers is transitive. As a step towards classifying Riemannian submersions with totally geodesic fibers, we consider the parameterized surface induced by lifting a base geodesic to po…
Classifies geodesic vectors in low-dimensional Lie algebras.
problem Stability of geodesic vectors in Lie algebras.
method Complete classification of Lyapunov stable and unstable geodesic vectors.
result Classification for metric Lie algebras of dimension 3 and 4.
We continue the study of the distribution of closed geodesics on nilmanifolds constructed from a simply connected 2-step nilpotent Lie group with a left invariant metric and a lattice. We consider a Lie group with an associated 2-step nilpotent Lie algebra constructed from an irreducible representation of a compact sem…
The paper studies algebraic relations of first integrals on specific Lie groups.
problem Algebraic relations of first integrals on step-two and step-three nilpotent Lie groups.
method Analysis of isometry algebra and invariant first integrals.
result Complete families of first integrals can be constructed with Killing vector fields and symmetric Killing 2-tensor fields in low dimensions.
Study magnetic geodesics on half-Lie groups, proving Hopf-Rinow theorem for energies above critical value.
problem Investigate magnetic geodesics on half-Lie groups using Riemannian and two-form structures.
method Define Mañé's critical value, prove Finsler geodesic flow equivalence, and apply Hopf-Rinow theorem.
result Hopf-Rinow theorem holds for energies above Mañé's critical value on magnetic geodesics.
Study on geodesics in Cartan group sub-Riemannian problem, proving conjugate time relation to Maxwell time.
problem Geodesics in sub-Riemannian problem on Cartan group.
method Analysis of symmetries, geodesic optimality, conjugate time calculation.
result First conjugate time is not less than Maxwell time, and equal for certain geodesics.
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 integrability of geodesic flow on specific Lie groups.
problem Integrability of geodesic flow on metabelian nilpotent groups.
method Symplectic reduction procedure applied to sub-Riemannian geodesic flow on metabelian nilpotent groups.
result Showed integrability of normal Hamiltonian flow in Engel-type groups.
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.
In this article we study the Hofer geometry of a compact Lie group K which acts by Hamiltonian diffeomorphisms on a symplectic manifold M. Generalized Hofer norms on the Lie algebra of K are introduced and analyzed with tools from group invariant convex geometry, functional and matrix analysis. Several global res…
Study geodesics on nested non-holonomic systems.
problem Interplay between geodesics on related non-holonomic systems.
method Hamiltonian formalism and geometric preliminaries.
result Presented several geometric examples, including exotic spheres and twistor space.
We prove injectivity and a support theorem for the X-ray transform on 2-step nilpotent Lie groups with many totally geodesic 2-dimensional flats. The result follows from a general reduction principle for manifolds with uniformly escaping geodesics.
A metric Lie algebra g is a Lie algebra equipped with an inner product. A subalgebra h of a metric Lie algebra g is said to be totally geodesic if the Lie subgroup corresponding to h is a totally geodesic submanifold relative to the left-invariant Riemannian metric defined by the inner product, on the simply connected …
In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogen…
The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterize by the property that any geodesic is an orbit of a 1-parameter group of isometries. In particular, we discuss some important totally geodesic submanifolds that inherit the property to be geodesic orbit. For a given geodesic…
In this paper, we study the set of homogeneous geodesics of a leftinvariant Finsler metric on Lie groups. We first give a simple criterion that characterizes geodesic vectors. As an application, we study some geometric properties of bi-invariant Finsler metrics on Lie groups. In particular a necessary and sufficient co…
The authors found geodesics, shortest arcs, cut loci, and conjugate sets for left-invariant sub-Riemannian matric on the Lie group SL(2), which is right-invariant relative to the Lie subgroup SO(2)⊂SL(2) (in other words, for invariant sub-Riemannian metric on weakly symmetric space $(SL(2)\times SO(2))/SO(2)…
The goals of this article are twofold : 1) to compute the conjugate locus of a geodesic that lies in the center of a simply connected, 2-step nilpotent Lie group with a left invariant metric 2) compare the isometry types of two such nilpotent Lie groups whose conjugate loci for central geodesics are "the same" in a sui…
The paper constructs a new Ricci-flat metric on almost abelian Lie groups.
problem Finding Lorentzian homogeneous Ricci-flat metrics on almost abelian Lie groups.
method Constructing left-invariant metrics on almost abelian Lie groups, focusing on dimensions four or higher.
result A new Ricci-flat metric that generalizes the Petrov solution to higher dimensions.
Normal geodesic flows flows of Carnot-Caratheodory are discussed from the point of view of the theory of Hamiltonian systems. The geodesic flows corresponding to left-invariant metrics and left- and -right-invariant rank 2 distributions on the three-dimensional Heisenberg group are analysed as integrable systems. The f…
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.
Study finds homogeneous spaces with geodesic orbits but no integrable distributions.
problem Characterizing homogeneous spaces with specific geometric properties.
method Examined Lie groups with compact stabilizers and classified spaces based on geodesic orbits and integrable distributions.
result Identified homogeneous spaces with geodesic orbits but lacking integrable invariant distributions.