The paper extends geodesic orbit sphere classification to Finsler geometry.
problem Classifying geodesic orbit spheres in Finsler geometry.
method Generalized from Riemannian to Finsler geometry, proving constant curvature conditions.
result Geodesic orbit Finsler spheres with constant flag curvature are Randers.
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.
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 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.
The paper studies Finsler spheres with constant flag curvature and finite orbits of prime closed geodesics.
problem Investigating Finsler spheres with specific curvature properties and geodesic orbits.
method Analyzing the action of isometries and loops on Finsler spheres, focusing on finite orbits of prime closed geodesics.
result The existence of geometrically distinct orbits of prime closed geodesics and their properties.
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.
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.
Homogeneous Finsler spheres with constant curvature have specific geodesic properties.
problem Existence and properties of homogeneous Finsler spheres with constant flag curvature.
method Proofs and analysis of geodesic properties on homogeneous Finsler spheres.
result Homogeneous Finsler spheres with constant flag curvature are either Riemannian or Randers.
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. We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the existence of a particular coding with two letters for the geodesics on these orbifolds.
New orbits found in Lagrangian systems on surfaces.
problem Finding action minimizing periodic orbits in Tonelli Lagrangian systems.
method Analyzing minimal boundaries and using graph theorems.
result Existence of action minimizing simple periodic orbits.
Study magnetic geodesic flows on spheres, describing their bifurcations.
problem Analyzing magnetic geodesic flows on 2-spheres.
method Generic pair of functions (f,Λ), Liouville fibration, Fomenko-Zieschang invariant, bifurcation diagrams. result Bifurcation diagrams consist of two curves in the (h,k)-plane. We recognize the Gromoll-Meyer sphere Sigma^7 as the geodesic join of a simple closed geodesic and a minimal subsphere Sigma^5, which can be equivariantly identified with the Brieskorn sphere W^5_3. As applications we in particular determine the full isometry group of Sigma^7, classify all closed subgroups that act fre…
The study finds infinitely many periodic orbits just above a critical value on a 2-sphere.
problem Finding periodic orbits just above a critical value on a 2-sphere.
method Introduced a new critical value c∞(L) and showed its strict inequality to the Mañé critical value c(L), proving the existence of infinitely many periodic orbits on energy levels e∈(c(L),c∞(L)). result Infinitely many periodic orbits exist on energy levels just above the Mañé critical value.
New invariants help find closed geodesics on curved spaces.
problem Finding closed geodesics on curved spaces.
method Constructing numerical homotopy invariants and applying them to functions on loop and sphere spaces.
result Proved new existence results for closed geodesics on Finsler manifolds.
Geodesic orbit Riemannian structures on R^n characterized.
problem Characterizing Riemannian manifolds with geodesic orbits.
method Geometric and algebraic characterization of geodesic orbit manifolds diffeomorphic to R^n.
result Established structural properties of geodesic orbit manifolds.
We consider the geodesic flow of reversible Finsler metrics on the 2-sphere and the 2-torus, whose geodesic flow has vanishing topological entropy. Following a construction of A. Katok, we discuss examples of Finsler metrics on both surfaces, which have large ergodic components for the geodesic flow in the unit tangent…
The study characterizes geodesic orbit Riemannian spaces and their properties.
problem Characterizing geodesic orbit Riemannian spaces and their properties.
method Analyzing the structure of nilradical and radical of Lie algebra, discussing compact Lie group representations.
result Described the structure of nilradical and radical of Lie algebra of isometry group.
Study classifies compact geodesic orbit spaces with two isotropy components.
problem Characterizing geodesic orbit Riemannian spaces.
method Classification of spaces with specific isotropy properties.
result Classification of compact geodesic orbit spaces with two isotropy summands.
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.
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.
The study finds geodesic orbit metrics on Lie groups from flag manifolds.
problem Investigating geodesic orbit metrics on Lie groups.
method Using generalized flag manifolds to form metrics on simple Lie groups.
result All left-invariant geodesic orbit metrics on simple Lie groups are naturally reductive.
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.
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 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.
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.
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.
The paper studies geodesic orbit Finsler metrics on Euclidean spaces.
problem Understanding Finsler metrics on Euclidean spaces with specific geometric properties.
method Analyzing Finsler metrics on Euclidean spaces to show they are fiber bundles over symmetric spaces.
result Finsler metrics on Euclidean spaces are fiber bundles over symmetric spaces with specific geometric properties.
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.
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.
New Einstein metric on G2 is found that's not geodesic orbit.
problem Finding left-invariant Einstein metrics on Lie groups that are not geodesic orbit.
method Developed tools for geodesic orbit Riemannian manifolds; used recent results by I. Chrysikos and Y. Sakane.
result Compact Lie group G2 admits a left-invariant Einstein metric that is not 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.
Geodesic orbit nilmanifolds linked to graph structures.
problem Characterizing Riemannian nilmanifolds associated with graphs.
method Analyzing naturally reductive and geodesic orbit properties of nilmanifolds defined by graphs.
result Nilmanifolds are geodesic orbit if and only if naturally reductive if and only if the defining graph is a disjoint union of complete graphs.
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 geodesic orbits on noncompact curved spaces, proving their distribution and counting.
problem Counting and equidistribution of periodic orbits on noncompact manifolds.
method Proved equidistribution in narrow topology, deduced exact asymptotic counting.
result Exact asymptotic counting of periodic orbits on noncompact manifolds.
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.
Study on periodic orbits in contact geometry and Finsler geodesics.
problem Existence and non-existence of periodic Reeb orbits on contact manifolds.
method Interpretation of Finsler geodesics as Reeb flows and use of open books.
result Existence of periodic Reeb orbits on contact manifolds with suitable open books.
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.
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.
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 $\SU(5)/\s(\U(2) imes \U(2))$.
Classifies finite orbits of mapping class group action on character varieties.
problem Classifying finite orbits of mapping class group action on character varieties of punctured spheres.
method Inductive proof using Lisovyy--Tykhyy's classification for 4-punctured spheres as base case.
result Proves no finite orbits for 7-punctured spheres and unique 1-parameter family for 6-punctured spheres.
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). 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.
The study finds sufficient conditions for Reeb flows to have genus zero global surfaces of section.
problem Finding conditions for Reeb flows to have genus zero global surfaces of section.
method Analyzes linking assumptions on periodic orbits and ambient contact geometry.
result Reveals sufficient conditions for Reeb flows to have genus zero global surfaces of section.
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 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…