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 reveals structure of isometry group for specific manifolds.
problem Understanding the isometry group of non-compact, homogeneous manifolds.
method Analyzes non-compact, homogeneous manifolds with immortal Ricci flows.
result Establishes structure result for isometry group.
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.
Free groups can be end homogeneity groups of 3-manifolds.
problem Tackling the possibility of free groups as end homogeneity groups of 3-manifolds.
method Constructing specific 3-manifolds with end homogeneity groups isomorphic to free groups.
result For every finitely generated free group, there exists an irreducible open 3-manifold with that group as its end homogeneity group.
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.
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.
Study characterizes k-rectifiable sets in homogeneous groups.
problem Characterizing k-rectifiable sets in arbitrary homogeneous groups. method Proves characterizations using (k,G)-approximate tangent groups. result Existence of (k,G)-approximate tangent groups implies k-rectifiability. Calculates affine transformations for specific homogeneous spaces.
problem Computing groups of affine transformations on homogeneous spaces.
method Analyzes conditions for affine connections and uses them to establish group isomorphisms.
result Groups of affine transformations are locally isomorphic under specified conditions.
Study new symmetries in non-symmetric spaces and discontinuous groups.
problem Analyze symmetries in non-symmetric homogeneous spaces and discontinuous groups.
method Investigate discrete series, discontinuous groups, and analysis on pseudo-Riemannian spaces.
result New insights into symmetries of non-symmetric homogeneous spaces and discontinuous groups.
A very important class of homogeneous Riemannian manifolds are the so-called normal homogeneous spaces, which have associated a canonical connection. In this work we obtain geometrically the (connected component of the) group of affine transformations with respect to the canonical connection for a normal homogeneous sp…
For every finitely generated abelian group G, we construct an irreducible open 3-manifold MG whose end set is homeomorphic to a Cantor set and with end homogeneity group of MG isomorphic to G. The end homogeneity group is the group of self-homeomorphisms of the end set that extend to homeomorphisms of the 3-m…
We explore the class of triples (M, nabla, P) where M is a manifold, nabla is an affine connection in M and P is a G-structure in M. Inside this class there are infinitesimally homogeneous manifolds, characterized by having G-constant curvature, torsion and inner torsion. For each matrix Lie group G subgroup of GL(Rn) …
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.
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.
Let G/H be a contractible homogeneous Sasaki manifold. A compact locally homogeneous aspherical Sasaki manifold Γ\G/H is by definition a quotient of G/H by a discrete uniform subgroup Γ≤G. We show that a compact locally homogeneous aspherical Sasaki manifold is always quasi-regular, that is, $…
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…
Hyperbolic groups' infinite orbits spread evenly in spaces.
problem Equidistribution of hyperbolic groups in homogeneous spaces.
method Averaging measures along spheres in Cayley graphs converges to Haar measure.
result Infinite orbits of hyperbolic groups equidistribute in homogeneous spaces.
Classifies homogeneous hypersurfaces in specific 4D geometries.
problem Classifying homogeneous hypersurfaces in 4D Thurston geometries.
method Analyzing subalgebras of Lie algebras and isometry groups.
result Determined all homogeneous hypersurfaces up to ambient isometries.
Study homogeneous Lorentzian manifolds under reductive Lie groups, reducing descriptions to semisimple groups.
problem Characterize homogeneous Lorentzian manifolds under reductive Lie groups.
method Analyze manifolds M=G/L for connected reductive Lie groups G and reductive subgroup L; focus on totally reducible isotropy representations. result Homogeneous Lorentzian manifolds reduce to semisimple Lie groups, and are reductive.
Symplectic manifolds which are homogeneous spaces of Poisson-Lie groups are studied in this paper. We show that these spaces are, under certain assumptions, covering spaces of dressing orbits of the Poisson-Lie groups which act on them. The effect of the Poisson induction procedure on such spaces is also examined, thus…
We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension n≥2 whose holomorphic automorphism group has dimension n2−2. This result complements an existing classification for automorphism group dimension n2−1 and greater obtained without the homogeneity assumption.
This paper constructs quandles with abelian inner automorphism groups from graphs, proving their homogeneity.
problem Finding quandles with specific automorphism properties.
method Starting from simple graphs, the paper constructs quandles with abelian inner automorphism groups and proves their homogeneity.
result Homogeneous quandles with abelian inner automorphism groups are constructed from vertex-transitive graphs.
Develops theory of homogeneous statistical manifolds and classifies Lie groups.
problem Understanding statistical manifolds and Lie groups.
method Constructs examples and classifies Lie groups using information geometry.
result Explicit examples of homogeneous statistical manifolds of low dimension constructed.
Study K-theory of homogeneous spaces of Lie groups.
problem Classifying and understanding the topological invariants of homogeneous spaces.
method Apply K-theory to homogeneous spaces of simply connected, compact Lie groups.
result Detailed analysis of four symmetric spaces, revealing their topological properties.
Geometrically revisits and models homogeneous spaces of compact Lie group G2.
problem Classifying homogeneous reductive spaces of compact Lie group G2. method Geometrical approach to revisit and model the spaces.
result Explicit relations among geometric models of the spaces.
The paper extends Heintze-Kobayashi-Wolf theory to negatively curved homogeneous Finsler manifolds.
problem Understanding negatively curved homogeneous Finsler manifolds.
method Generalizing Heintze-Kobayashi-Wolf theory to homogeneous Finsler geometry, proving two main theorems.
result Negatively curved homogeneous Finsler manifolds are isometric to Lie groups with specific properties.
The paper studies the limits and rigidity of almost homogeneous spaces with Ricci curvature bounds.
problem Understanding the limits and rigidity of almost homogeneous spaces with Ricci curvature bounds.
method Analyzes sequences of almost homogeneous RCD(K,N) spaces and their Gromov-Hausdorff limits.
result The Gromov-Hausdorff limit of a sequence of almost homogeneous RCD(K,N) spaces is a nilpotent Lie group with Ric ≥ K.
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.
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.
Classifies 1-connected Lorentzian manifolds with essential conformal groups.
problem Classifying Lorentzian manifolds with specific conformal groups.
method Proves existence of a metric making the manifold homogeneous and plane wave.
result Completes the classification of 1-connected Lorentzian manifolds with transitive conformal groups.
Study how algebraic conditions on isotropy group affect Lorentzian homogeneous space geometry.
problem Understand how algebraic conditions on isotropy group affect the geometry and curvature of Lorentzian homogeneous spaces.
method Prove that a Lorentzian locally homogeneous space is locally isometric to a plane wave if it admits an Ambrose--Singer connection with indecomposable, non-irreducible holonomy.
result Generalize existing results about Lorentzian homogeneous spaces with irreducible isotropy and prove results about Lorentzian connections with parallel torsion and 2-symmetric connections.
Bi-invariant metrics on Lie groups and homogeneous spaces are extremal and rigid.
problem Gromov's question on extremality of bi-invariant metrics on compact Lie groups.
method Proving rigidity of bi-invariant metrics on compact Lie groups and homogeneous spaces.
result Bi-invariant metrics on compact Lie groups and homogeneous spaces are extremal and rigid.
We describe the structure of d-dimensional homogeneous Lorentzian G-manifolds M=G/H of a semisimple Lie group G. Due to a result by N. Kowalsky, it is sufficient to consider the case when the group G acts properly, that is the stabilizer H is compact. Then any homogeneous space G/Hˉ with a smaller gro…
Study index theory on Lie group homogeneous spaces using topological and analytic methods.
problem Index theory on homogeneous spaces of Lie groups.
method Topological and analytic approaches: Riemann-Roch formula and heat kernel methods.
result Local index formula representing higher indices of equivariant elliptic operators.
In this article, we study Einstein Kropina metrics on Lie groups and homogeneous spaces. We give a method to construct Einstein Kropina metrics on Lie groups. As an example of this method, a family of non-Riemannian Einstein Kropina metrics on the special orthogonal group SO(n) is given. Then, we classify all left in…
The notions of \emph{Poisson Lie group} and \emph{Poisson homogeneous space} are extended to the Dirac category. The theorem of Drinfel′d (\cite{Drinfeld93}) on the one-to-one correspondence between Poisson homogeneous spaces of a Poisson Lie group and a special class of Lagrangian subalgebras of the Lie bialgebra as…
Let (M,F) be a connected Finsler space. An isometry of (M,F) is called a Clifford-Wolf translation (or simply CW-translation) if it moves all points the same distance. The compact Finsler space (M,F) is called restrictively Clifford-Wolf homogeneous (restrictively CW-homogeneous) if for any two sufficiently close…
Abstract: Characterizes spaces with positive scalar curvature.
problem Spaces with positive scalar curvature and invariant metrics.
method Cohomogeneity one manifolds and homogeneous spaces with compact Lie group actions.
result Characterization of spaces with positive scalar curvature.
The paper generalizes cyclic metrics in homogeneous Finsler geometry.
problem Understanding cyclic metrics in homogeneous Finsler spaces.
method Generalization of cyclic metrics, proving conditions for symmetry, and constructing cyclic metrics.
result A Finsler cyclic Lie group with an Abelian Lie algebra.
We classify all compact simply connected homogeneous CR manifolds M of codimension one and with non-degenerate Levi form up to CR equivalence. The classification is based on our previous results and on a description of the maximal connected compact group G(M) of automorphisms of M. We characterize also the standa…
We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we show that there do exist non-compact and non-complete examples, where the linear ho…
In the vein of Bonfert-Taylor, Bridgeman, Canary, and Taylor we introduce the notion of quasiconformal homogeneity for closed oriented hyperbolic surfaces restricted to subgroups of the mapping class group. We find uniform lower bounds for the associated quasiconformal homogeneity constants across all closed hyperbolic…
The paper explores conformal groups on plane waves and proves a conjecture in locally homogeneous settings.
problem Proving the Lorentzian conformal Lichnerowicz conjecture in locally homogeneous settings.
method Analyzing conformal groups on plane waves and proving the conjecture in a specific setting.
result The Lorentzian conformal Lichnerowicz conjecture is proven in a locally homogeneous setting.
The main result of this paper is the classification of the real irreducible representations of compact Lie groups with vanishing homogeneity rank.
Characterizes quandles with abelian inner automorphisms.
problem Understanding quandles with specific automorphism properties.
method Generalizes previous work to construct new quandles.
result Homogeneous quandles with abelian inner automorphisms are abelian extensions of trivial quandles.
Study of special Lorentzian Lie groups with 4D isometry group, finding all are non-gradient expanding Ricci solitons.
problem Characterizing homogeneous Lorentzian three-manifolds with a 4D isometry group.
method Explicit global coordinate description and proof of Ricci soliton properties.
result All special examples are non-gradient expanding Ricci solitons.
Let G be a complex reductive connected algebraic group equipped with the Sklyanin bracket. A classification of Poisson homogeneous G-spaces with connected isotropy subgroups is given. This result is based on Drinfeld's correspondence between Poisson homogeneous G-spaces and Lagrangian subalgebras in the double $D…
We prove various classification results for homogeneous locally conformally symplectic manifolds. In particular, we show that a homogeneous locally conformally Kaehler manifold of a reductive group is of Vaisman type, if the normalizer of the isotropy group is compact. We also show that such a result does not hold in t…