Introduces new types of homogeneous spaces and their properties.
problem Defining and understanding new types of homogeneous spaces.
method Introducing and analyzing (strongly) (Θ-)discrete homogeneous spaces. result Discovers relationships between new and existing homogeneous space types.
The study finds homogeneous geodesics in homogeneous Kropina spaces.
problem Existence of homogeneous geodesics in homogeneous Kropina spaces.
method Analyzes homogeneous Kropina spaces and (α,β)-homogeneous spaces to find geodesics. result Homogeneous Kropina spaces admit at least one homogeneous geodesic through any point.
Study finds six homogeneous surfaces with multiple invariant connections.
problem Characterizing homogeneous surfaces with invariant connections.
method Computed all simply connected homogeneous and infinitesimally homogeneous surfaces.
result Found six non-equivalent surfaces with multiple invariant connections.
Introduces homogeneity supermanifolds for studying graded structures.
problem Graded structures on supermanifolds.
method Homogeneity degrees and weight vector field.
result Proofs of homogeneous Poincaré Lemma, Frobenius Theorem, and Darboux Theorem.
Homogeneous three-spheres have only homogenous foliations.
problem Characterize foliations of homogeneous three-spheres.
method Prove that a three-sphere's metric foliations are homogenous if and only if it is naturally reductive.
result Homogeneous three-spheres have only homogenous foliations.
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…
Corrected proof of geodesics in homogeneous Finsler spaces using affine method.
problem Proving the existence of homogeneous geodesics in homogeneous Finsler spaces.
method Used affine method to correct the proof of the algebraic method.
result Correctly proved that any homogeneous Finsler space admits a homogeneous geodesic through any point.
We prove that the Penrose limit of a spacetime along a homogeneous geodesic is a homogeneous plane wave spacetime and that the Penrose limit of a reductive homogeneous spacetime along a homogeneous geodesic is a Cahen--Wallach space. We then consider several homogenous examples to show that these results are indeed sha…
We prove that under some purely algebraic conditions every locally homogeneous structure modelled on some homogeneous space is induced by a locally homogeneous structure modelled on a different homogeneous space.
Our purpose is to use a Darboux homogenous derivative to understand the harmonic maps with values in homogeneous space. We present a characterization of these harmonic maps from the geometry of homogeneous space. Furthermore, our work covers all type of invariant geometry in homogeneous space.
We show that a Lorentzian homogeneous space admitting a homogeneous structure of type T1 + T3 is either a (locally) symmetric space or a singular homogeneous plane wave.
The study verifies a conjecture about homogeneous quotients of manifolds with positive curvature.
problem Verifying the Homogeneity Conjecture for manifolds with positive curvature.
method Examining globally homogeneous Riemannian quotients of homogeneous manifolds, focusing on those admitting a positive curvature metric.
result Further evidence supports the Homogeneity Conjecture for certain manifolds with positive curvature.
In this article we study homogeneous warped product Einstein metrics and its connections with homogeneous Ricci solitons. We show that homogeneous (λ,n+m)-Einstein manifolds (which are the bases of homogeneous warped product Einstein metrics) are one-dimensional extensions of algebraic solitons. This answers a questi…
The Homogeneity Conjecture explores if constant displacement isometries imply homogeneous spaces.
problem Whether constant displacement isometries on a manifold imply its homogeneity.
method Survey and verification of cases, including new results.
result New results and open problems suggested.
In previous papers, a fundamental affine method for studying homogeneous geodesics was developed. Using this method and elementary differential topology it was proved that any homogeneous affine manifold and in particular any homogeneous pseudo-Riemannian manifold admits a homogeneous geodesic through arbitrary point. …
Survey of recent results on homogeneous finite-dimensional spaces.
problem Understanding properties of homogeneous finite-dimensional spaces.
method Discussion of recent results and unsolved problems.
result Discussion of recent results and unsolved problems.
The paper extends two-step homogeneous geodesics to homogeneous Finsler spaces.
problem Extending two-step homogeneous geodesics to Finsler spaces.
method Providing sufficient conditions for (α,β) spaces and decomposable cubic spaces to have two-step Finsler geodesic orbit spaces. result Presented examples of two-step Finsler geodesic orbit spaces.
Smooth manifolds from locally homogeneous spaces.
problem Understanding smoothness in C0-Riemannian manifolds. method Demonstrated locally homogeneous C0-Riemannian manifolds are smooth. result Locally homogeneous C0-Riemannian manifolds are smooth. We study a family of 3-dimensional Lorentz manifolds. Some members of the family are 0-curvature homogeneous, 1-affine curvature homogeneous, but not 1-curvature homogeneous. Some are 1-curvature homogeneous but not 2-curvature homogeneous. All are 0-modeled on indecomposible local symmetric spaces. Some of the members…
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 locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection on a two dimensional torus is complete and, up to a finite cover, homogeneous. Let ∇ be …
We extend the definition of curvature homogeneity of type (1,3) to include the possibility that there is a homothety between any two points of a manifold preserving the first r covariant derivatives of the curvature operator simultaneously; we call this strong curvature homogeneity of type (1,3) up to order r. We chara…
Paper classifies positively curved δ-homogeneous Finsler metrics.
problem Classifying positively curved Finsler metrics.
method Exploring δ-homogeneity and using approximation techniques.
result Classification of positively curved δ-homogeneous Finsler spaces.
The paper classifies compact homogeneous Finsler manifolds with positive flag curvature.
problem Classifying compact homogeneous Finsler manifolds with positive flag curvature.
method Defined and classified very standard homogeneous Finsler metrics on compact homogeneous Lie groups.
result Classified all compact homogeneous Lie groups admitting positively curved very standard homogeneous Finsler metrics.
The paper studies Randers and (α,β) equigeodesics on compact homogeneous manifolds.
problem Characterizing equigeodesics on compact homogeneous manifolds.
method Analyzing different types of equigeodesics (Riemannian, Finsler, Randers, (α,β)) on compact homogeneous manifolds. result Randers and (α,β) equigeodesics are equivalent on compact homogeneous manifolds, and a criterion is found. Researchers found all homogeneous structure tensors on two specific 3D manifolds.
problem Classifying homogeneous structure tensors on specific 3D manifolds.
method Determined all homogeneous structure tensors on S2imesR and H2imesR. result Complete classification of homogeneous structure tensors on three-dimensional homogeneous Riemannian manifolds.
Study verifies Homogeneity Conjecture for three odd-dimensional spheres in positive curvature.
problem Verifying the Homogeneity Conjecture for three specific odd-dimensional spheres in positive curvature.
method Developed methods to verify the conjecture for three odd-dimensional spheres.
result Completes verification of the Homogeneity Conjecture in positive curvature.
Study how regularization and optimization affect margin in deep models.
problem Understanding margin maximization in deep learning models.
method Analyze the limit of loss minimization with diverging norm constraints and margin paths.
result Discovers lexicographic max-margin solutions for homogeneous models and shows convergence under certain conditions.
New tools compute index of symmetry in homogeneous fibrations.
problem Computing the index of symmetry in homogeneous fibrations.
method Developed new tools to compute the index of symmetry.
result Determined the index of symmetry of various homogeneous spaces.
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 shows magnetic trajectories in Berger spheres are homogeneous.
problem Homogeneity of contact magnetic trajectories in Berger spheres.
method Proved every contact magnetic trajectory is a product of a homogeneous geodesic and a charged Reeb flow.
result Contact magnetic trajectories in Berger spheres are homogeneous.
Classifies special homogeneous curves with polynomial equations.
problem Identifying and classifying special homogeneous curves.
method Analyzing homogeneous polynomials and their level sets with group actions.
result All special homogeneous curves are classified.
Calculates S-curvature and mean Berwald curvature in homogeneous Finsler spaces.
problem Calculating curvatures in homogeneous Finsler spaces.
method Explicit formula for S-curvature and mean Berwald curvature in homogeneous generalized m-Kropina metric.
result Explicit formula for S-curvature and mean Berwald curvature in homogeneous generalized m-Kropina metric.
Improves arc separation result for homogeneous spaces.
problem Separating regions in homogeneous spaces by arcs.
method Using homogeneity instead of strong local homogeneity, and considering arcs with one interior point.
result Regions in homogeneous spaces of dimension ≥ 2 are not separated by arcs.
We will discuss in this paper homogeneous locally conformally Keahler (or shortly homogeneous l.c.K.) manifolds and locally homogeneous l.c.K. manifolds from various aspects of study in the field of l.c.K. geometry. We will provide a survey of known results along with some new results and observations; in particular we…
In this paper, we study normal homogeneous Finsler spaces. We first define the notion of a normal homogeneous Finsler space, using the method of isometric submersion of Finsler metrics. Then we study the geometric properties. In particular, we establish a technique to reduce the classification of normal homogeneous Fin…
Homogeneous quasipositive links are positive if their Seifert circles match braid index.
problem When are homogeneous quasipositive links positive?
method Showed that a specific type of diagram is positive.
result Homogeneous diagrams with matching Seifert circles and braid index are positive.
We study in this paper previously defined by V.N. Berestovskii and C.P. Plaut δ-homogeneous spaces in the case of Riemannian manifolds. Every such manifold has non-negative sectional curvature. The universal covering of any δ-homogeneous Riemannian manifolds is itself δ-homogeneous. In turn, every simply connecte…
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.
Homogeneous magnetic trajectories in a special linear group proven.
problem Proving homogeneity of magnetic trajectories in a specific group.
method Using contact magnetic curves and geodesics.
result Every contact magnetic trajectory is a product of a homogeneous geodesic and a charged Reeb flow.
Study spinor flow on homogeneous spin manifolds.
problem Understanding spinor flow on specific homogeneous manifolds.
method General setup provided; detailed analysis in dimensions 3 and 6.
result Detailed analysis of spinor flow on specific examples.
Study on Ricci flows of awesome homogeneous spaces, proving finite extinction time.
problem Understanding the long-time behavior of Ricci flows on homogeneous spaces.
method Analyzing Ricci flows on non-compact manifolds, focusing on finite extinction time.
result Ricci flows on non-contractible spaces have finite extinction time, confirming conjecture.
We explore the plane-wave limit of homogeneous spacetimes. For plane-wave limits along homogeneous geodesics the limit is known to be homogeneous and we exhibit the limiting metric in terms of Lie algebraic data. This simplifies many calculations and we illustrate this with several examples. We also investigate the beh…
Paper proves every homogeneous Landsberg surface is either Riemannian or locally Minkowskian.
problem Characterizing homogeneous Landsberg surfaces.
method Proved isotropic flag curvature and used it to prove rigidity.
result Every homogeneous Landsberg surface is either Riemannian or locally Minkowskian.
Researchers analyze charged particle motion in specific homogeneous spaces.
problem Understanding charged particle motion in homogeneous spaces.
method Examined motion under natural electromagnetic fields in compact homogeneous spaces.
result Described motion in homogeneous Riemannian spaces with various deformations.
Classifies polar actions on 3D homogeneous spaces.
problem Classifying polar isometric actions on 3D homogeneous spaces.
method Orbit equivalence classification and study of cohomogeneity one actions.
result Classification of extrinsically homogeneous surfaces and orbit foliations.
In this paper, we introduce a study of prolongations of homogeneous vector bundles. We give an alternative approach for the prolongation. For a given homogeneous vector bundle E, we obtain a new homogeneous vector bundle. The homogeneous structure and its corresponding representation are derived. The prolongation of in…
Homogeneous magnetic paths found in Heisenberg space.
problem Understanding magnetic geodesics in the Heisenberg group.
method Proving homogeneity of magnetic geodesics derived from the canonical contact structure.
result Magnetic geodesics in the Heisenberg group are homogeneous.