Geodesic orbit and weakly symmetric properties in spray geometry.
problem Understanding properties of homogeneous spray manifolds.
method Using reductive decompositions and spray vector fields to describe and prove properties.
result Weakly symmetric spray manifolds are geodesic orbit manifolds.
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…
A geodesic orbit manifold is a complete Riemannian manifold all of whose geodesics are orbits of one-parameter groups of isometries. We give both a geometric and an algebraic characterization of geodesic orbit manifolds that are diffeomorphic to Rn. Along the way, we establish various structural properties …
Study on geodesics of Finsler metrics derived from Riemannian metrics.
problem Investigating geodesics in Finsler metrics derived from Riemannian metrics.
method Proved geodesic lemma for a family of Riemannian geodesic orbit metrics, analyzed geodesic graphs.
result Derived Finsler metrics have geodesic orbit property and belong to a new class of metrics.
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.
Classifies geodesic orbit spaces with abelian isotropy subgroups.
problem Characterizing and classifying geodesic orbit spaces with specific isotropy subgroups.
method Simplified study of geodesic orbit metrics on G/S by reducing to submanifolds and generalized flag manifolds, using properties of root systems.
result Geodesic orbit spaces of the form (G/S,g) are naturally reductive.
Geodesic orbit property studied for Lorentz manifolds.
problem Geodesic orbit property for Lorentz manifolds.
method Defined naturally reductive for pseudo-Riemannian manifolds and proved theorems for Lorentz nilmanifolds.
result Geodesic orbit property holds for Lorentz nilmanifolds under specific conditions.
Geodesic graphs for special Finsler metrics on spheres are studied.
problem Characterizing geodesic orbit Finsler metrics on spheres.
method Explicit constructions and group extensions.
result Not all projective spaces admit invariant Finsler metrics.
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. The main result is the classification of compact simply connected geodesic orbit Riemannian spaces G/H with two irreducible submodules in…
Geodesic orbit metrics on real flag manifolds identified.
problem Classifying real flag manifolds with geodesic orbit metrics.
method Investigated invariant metrics on real flag manifolds, focusing on those where geodesics are orbits of one-parameter subgroups.
result Non-trivial geodesic orbit metrics exist on real flag manifolds, unlike in the complex case.
Study geodesic orbit Lorentz nilmanifolds, proving structural properties.
problem Understanding the structure of geodesic orbit Lorentz nilmanifolds.
method Analyzing geodesic orbit Lorentz nilmanifolds with reductive decompositions.
result Proves properties of nilpotent subgroups and their nilpotency steps.
We study the geodesic orbit property for nilpotent Lie groups N when endowed with a pseudo-Riemannian left-invariant metric. We consider this property with respect to different groups acting by isometries. When N acts on itself by left-translations we show that it is a geodesic orbit space if and only if the metric…
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.
Researchers extend geodesic orbit properties to pseudo-Riemannian nilmanifolds of specific signature.
problem Extending geodesic orbit properties to pseudo-Riemannian manifolds of signature (n-2,2).
method Analyzing trans-Lorentz nilmanifolds with reductive decomposition and nilpotent subgroup.
result Characterization of nilpotent subgroups and structure of nilmanifolds.
New surfaces with special geodesic and horocycle behaviors discovered.
problem Understanding geodesic and horocycle dynamics on hyperbolic surfaces.
method Constructing geometrically infinite hyperbolic surfaces with tailored recurrence properties.
result First examples of non-trivial minimal horocyclic orbit closures and infinite locally-finite conservative horocyclic invariant measures.
This survey explores compact geodesic orbit manifolds and their properties.
problem Classifying compact geodesic orbit manifolds.
method Review and analysis of existing results.
result Study of geodesic orbit condition for $\SU(5)/\s(\U(2) imes \U(2))$.
Geodesic orbit spaces and their families are studied in pseudo-Riemannian manifolds.
problem Understanding geodesic orbit spaces and their properties in pseudo-Riemannian manifolds.
method Analyzing real form families of pseudo-Riemannian manifolds and proving properties of geodesic orbit spaces.
result Geodesic orbit spaces and their families have interesting properties in pseudo-Riemannian manifolds.
The paper studies geodesic orbit properties in Finsler spaces.
problem Investigating geodesic orbit properties in homogeneous Finsler spaces.
method Introduced metric operator and defined standard homogeneous Finsler metrics.
result Classified homogeneous manifolds with specific geodesic orbit properties.
The paper classifies Sasaki-Einstein orbits in compact Hermitian symmetric spaces.
problem Classifying Sasaki-Einstein orbits in compact Hermitian symmetric spaces.
method Examining orbits as CR submanifolds, proving total geodesy, and analyzing contact structures.
result Completely determine Sasaki-Einstein orbits.
In this paper, we establish a sufficient condition for a geodesic in a Riemannian manifold to be homogeneous, i.e. an orbit of an 1-parameter isometry group. As an application of this result, we provide a new proof of the fact that every weakly symmetric space is geodesic orbit manifold, i.e. all its geodesics are ho…
New surgery method preserves Anosov flow properties using bi-contact geometry.
problem Defining a new type of surgery on Anosov flows.
method Bi-contact geometry and Reeb dynamics.
result Necessary and sufficient condition for generating contact Anosov flows.
The paper classifies geodesic orbit spaces with simple isotropy groups.
problem Classifying geodesic orbit spaces with simple isotropy groups.
method Classifying G-naturally reductive and G-geodesic orbit metrics on M. result Classification of geodesic orbit spaces with simple isotropy groups.
Study on homogeneous geodesics in sub-Riemannian geometry.
problem Characterizing and understanding homogeneous geodesics in sub-Riemannian manifolds.
method Criterion for geodesics to be homogeneous, proof of geodesic orbit spaces, examples of geodesic orbit sub-Riemannian manifolds.
result Existence of at least one homogeneous geodesic under broad conditions.
Study geodesic orbit metrics in quaternionic Stiefel manifolds.
problem Characterize geodesic orbit spaces in quaternionic Stiefel manifolds.
method Analyze homogeneous Riemannian spaces (M=G/H,g) with geodesics as orbits of subgroups. result Identify conditions for $(\Sp(n)/\Sp(n_1) imes \cdots imes \Sp(n_s), g)$ to be a geodesic orbit space.
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.
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.
Extends flat submanifold properties from hyperbolic plane to symmetric spaces.
problem Spectral asymptotics for orbital integrals in symmetric spaces.
method Generalizes geodesic properties to maximal flat submanifolds.
result Establishes geometric properties of maximal flat submanifolds in symmetric spaces.
A geodesic orbit manifold (GO manifold) is a Riemannian manifold (M,g) with the property that any geodesic in M is an orbit of a one-parameter subgroup of a group G of isometries of (M,g). The metric g is then called a G-GO metric in M. For an arbitrary compact homogeneous manifold M=G/H, we simplify the general proble…
Homoclinic orbits found in geodesic flows on surfaces.
problem Existence of homoclinic orbits in geodesic flows.
method Kupka-Smale metric on closed surfaces.
result Homoclinic orbits for all hyperbolic geodesics.
Geodesic orbit metrics proven on specific homogeneous spaces.
problem Geodesic orbit metrics on homogeneous spaces.
method Using strongly isotropy irreducible spaces and proving natural reductivity.
result Geodesic orbit metrics are naturally reductive on constructed homogeneous spaces.
Study geodesic orbit metrics on specific homogeneous spaces.
problem Characterize geodesic orbit spaces in a class of homogeneous bundles.
method Analyze geodesic orbit spaces of compact Lie groups with semisimple subgroups.
result Identify conditions for a metric to be geodesic orbit.
New families of non-singular geodesic orbit nilmanifolds discovered.
problem Classifying non-singular geodesic orbit nilmanifolds.
method Complete classification through analysis of nilmanifolds.
result New families of non-singular GO nilmanifolds with dimensions 14 and 15.
Study periodic geodesics on contact 3D manifolds, proving existence and precise properties.
problem Existence and properties of periodic geodesics in contact sub-Riemannian metrics.
method Develops two independent subjects: existence of spiraling geodesics and precise study of geodesics on quotient of SL2(R).
result Proves existence and precise properties of periodic geodesics.
The study bounds the number of closed geodesics in a specific orbit closure of surfaces.
problem Counting closed geodesics in a specific orbit closure of surfaces.
method Analyzes triangulations and Teichmüller geodesics to bound the number of closed geodesics.
result Obtains exponential bounds on the number of closed geodesics of length at most R.
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.
A Finsler space (M,F) is called a geodesic orbit space if any geodesic of constant speed is the orbit of a one-parameter subgroup of isometries of (M,F). In this paper, we study Finsler metrics on Euclidean spaces which are geodesic orbit metrics. We will show that, in this case (M,F) is a fiber bundle over a s…
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 geodesics on SL(n,R) adjoint orbits.
problem Geodesics on adjoint orbits of SL(n,R). method Translate problem into tangent bundle of SO(n)-flag manifolds, use Sasaki metric and Lie Theory. result Explicit description of geodesics families in SL(2,R). 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.
Geodesics spiral around Reeb orbits in 3D contact manifolds.
problem Understanding geodesics in sub-Riemannian geometry.
method Normal form along Reeb orbits due to Melrose.
result Sub-Riemannian geodesics spiral around Reeb orbits in both phase and configuration spaces.
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.
We establish a precise asymptotic formula for the number of homotopy classes of periodic orbits for the geodesic flow on rank one manifolds of nonpositive curvature. This extends a celebrated result of G. A. Margulis to the nonuniformly hyperbolic case and strengthens previous results by G. Knieper. We also establish s…
In this paper, we generalize the classification of geodesic orbit spheres from Riemannian geometry to Finsler geometry. Then we further prove if a geodesic orbit Finsler sphere has constant flag curvature, it must be Randers. It provides an alternative proof for the classification of invariant Finsler metrics with $K\e…
Proves a quantitative closing lemma for negatively curved manifolds.
problem Closing lemma for negatively curved manifolds.
method Quantitative closing lemma proof.
result Study of partner and pseudo-partner orbits for self-crossing closed geodesics.
We study cohomogeneity one Riemannian manifolds and we establish some simple criterium to test when a singular orbit is totally geodesic. As an application, we classify compact, positively curved Riemannian manifolds which are acted on isometrically by a non semisimple Lie group with an hypersurface orbit.
We prove some rigidity results on geodesic orbit Finsler spaces with non-positive curvature. In particular, we show that a geodesic Finsler space with strictly negative flag curvature must be a non-compact Riemannian symmetric space of rank one.
Counting geodesics on compact symmetric spaces using orbit dimensions and topological data.
problem Counting geodesics on compact symmetric spaces.
method Using orbit dimensions and topological data of the symmetric space.
result Obtained data on dimensions and connected components of focal orbits.
Study horocycle orbits in Z-covers of hyperbolic surfaces.
problem Classify horocycle orbit closures in Z-covers of compact hyperbolic surfaces. method Careful analysis of distance minimizing geodesic rays in the cover.
result All non-maximal horocycle orbit closures have integer Hausdorff dimension.