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.
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 equigeodesics on compact homogeneous spaces using Lie algebra properties.
problem Identifying equigeodesic vectors on compact homogeneous spaces.
method Formula for equigeodesic vectors based on isotropy representation and Lie algebra structure.
result Identification of equigeodesic vectors solely through Lie algebra properties.
Compact pseudo-Hermitian spaces have rigid holomorphic isometries.
problem Characterizing the rigidity of pseudo-Hermitian homogeneous spaces.
method Analysis of Tits fibration and automorphism groups of compact spaces.
result Holomorphic isometries of compact pseudo-Hermitian spaces are compact.
Study extends reflective submanifold theory to compact homogeneous spaces.
problem Characterize reflective submanifolds in compact isotropy irreducible spaces.
method Extend previous results to infinite-dimensional Hilbert spaces.
result Inverse image of reflective submanifolds is also reflective.
Existence and convergence of ancient Ricci flow solutions on compact homogeneous spaces.
problem Existence and characterization of ancient solutions to the Ricci flow on compact homogeneous spaces.
method General existence theorem and Gromov-Hausdorff convergence under rescaling.
result Convergence of collapsed ancient solutions to Einstein metrics on torus fibrations.
Study extends Kobayashi's method to non-reductive subgroups for homogeneous spaces.
problem Existence of compact Clifford-Klein forms in homogeneous spaces.
method Extend Kobayashi's method to non-reductive subgroups and compare Cartan projections and non-compact dimensions.
result Examples of homogeneous spaces without compact Clifford-Klein forms.
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.
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.
Study shows algebraic nature of manifold submetries on compact spaces.
problem Understanding manifold submetries on compact homogeneous spaces.
method Analyzes singular Riemannian foliations and manifold submetries on compact normal homogeneous spaces.
result Establishes a one-to-one correspondence between algebras of preserved functions and manifold submetries.
We study locally compact metric spaces that enjoy various forms of homogeneity with respect to Möbius self-homeomorphisms. We investigate connections between such homogeneity and the combination of isometric homogeneity with invertibility. In particular, we provide a new characterization of snowflakes of boundaries of …
Compact quotients of homogeneous spaces are studied, leading to new findings about sphere bundles.
problem Understanding compact quotients of reductive homogeneous spaces and their implications.
method Analyzing normal bundles and sphere bundles associated with these spaces, proving homotopy triviality conditions.
result Many reductive homogeneous spaces do not admit compact quotients, resolving conjectures.
We complete our recent classification of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank …
New infinite families of flat spaces found from symmetric spaces.
problem Finding new flat homogeneous spaces.
method Starting from compact symmetric spaces, constructing infinite families of compact homogeneous spaces with invariant Bismut connections.
result Infinite families of compact homogeneous spaces with vanishing Ricci tensor.
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…
The study classifies homogeneous manifolds with specific geometric properties.
problem Classifying homogeneous manifolds with Riemannian and Finsler equigeodesic properties.
method Analyzes homogeneous manifolds G/H and their decompositions into Euclidean and compact isotropy irreducible factors. result Classifies homogeneous manifolds into Riemannian and Finsler equigeodesic spaces.
Proves conditions for separating regions in homogeneous spaces without trivial topology.
problem Separating regions in homogeneous, locally compact spaces without trivial topology.
method Analyzes properties of closed subsets and their boundaries in Čech cohomology.
result Conditions for irreducible separation without trivial topology.
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.
In this note we classify all homogeneous spaces G/H admitting a G-invariant G2-structure, assuming that G is a compact Lie group and G acts effectively on G/H. They include a subclass of all homogeneous spaces G/H with a G-invariant G~2-structure, where G is a compact Lie group. There are ma…
The author reviews his results on locally compact homogeneous spaces with inner metric, in particular, homogeneous manifolds with inner metric. The latter are isometric to homogeneous (sub-)Finslerian manifolds; under some additional conditions they are isometric to homogeneous (sub)-Riemannian manifolds. The class Ω…
Ancient Ricci flows on compact spaces converge to solitons.
problem Understanding long-time behavior of Ricci flows on compact spaces.
method Proving precompactness of invariant metrics and analyzing blow-down sequences.
result Ancient homogeneous Ricci flows on compact manifolds converge to gradient shrinking solitons.
We classify compact homogeneous geometries of irreducible spherical type and rank at least 2 which admit a transitive action of a compact connected group, up to equivariant 2-coverings. We apply our classification to polar actions on compact symmetric spaces.
This article deals with fat bundles. Berard-Bergery classified all homogeneous bundles of that type. We ask a question of a possibility to generalize his description in the case of arbitrary G-structures over homogeneous spaces. We obtain necessary conditions for the existence of such bundles. These conditions yield a …
Study on metrics and eigenvalues for compact homogeneous spaces.
problem Estimate the Laplace eigenvalue for compact homogeneous Riemannian manifolds.
method Investigate the functional g↦λ1(G/K,g)diam(G/K,g)2 for compact homogeneous spaces G/K. result Prove the existence of an upper bound for the mentioned functional for all compact homogeneous spaces with multiplicity-free isotropy representation.
Spaces are classified as almost homology n-manifolds if their homology groups are trivial for all but the last dimension.
problem Classifying spaces as almost homology n-manifolds.
method Providing a necessary and sufficient condition for locally compact homogeneous ANR-spaces or strongly locally homogeneous ANR-spaces to be almost homology n-manifolds.
result Spaces are classified as almost homology n-manifolds based on their homology groups.
This paper proves that there are no compact forms for a large class of homogeneous spaces admitting actions by higher-rank semisimple Lie groups. It builds on Zimmer's approach for studying such spaces using cocycle superrigidity. The proof involves cocycle superrigidity, measure rigidity for unipotent flows, technique…
This article studies the volume of compact quotients of reductive homogeneous spaces. Let G/H be a reductive homogeneous space and Γ a discrete subgroup of G acting properly discontinuously and cocompactly on G/H. We prove that the volume of Γ\G/H is the integral, over a certain homology class of $Γ…
In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special li…
Graev's nerve implies invariant Einstein metrics on homogeneous spaces.
problem Existence of invariant Einstein metrics on homogeneous spaces.
method Lie-theoretic definition of Graev's nerve and curvature estimates.
result Detailed description of Graev's work and curvature estimates.
New homogeneous manifolds with invariant Bismut Ricci flat connections are constructed.
problem Constructing homogeneous manifolds with invariant Bismut Ricci flat connections.
method Classification and construction of homogeneous spaces with specific properties.
result Examples of compact homogeneous Riemannian manifolds with invariant Bismut Ricci flat connections are provided.
Strong formal properties for toric and homogeneous Kähler manifolds.
problem Understanding formal properties of Kähler manifolds.
method Analyzing rationally and strongly formal properties of toric and homogeneous Kähler manifolds.
result Toric and homogeneous Kähler manifolds are both rationally and strongly formal.
Study the stability of Einstein metrics on homogeneous spaces.
problem Classify Einstein metrics on homogeneous spaces.
method Analyze the scalar curvature functional to understand stability.
result Identify the nature of each Einstein metric as a critical point.
We develop a general structure theory for compact homogeneous Riemannian manifolds in relation to the co-index of symmetry. We will then use these results to classify irreducible, simply connected, compact homogeneous Riemannian manifolds whose co-index of symmetry is less or equal than three. We will also construct ma…
Study finds only two CROSSes can have specific quaternionic structure.
problem Characterizing compact rank one symmetric spaces with quaternionic structures.
method Analyzing properties of CROSSes (compact rank one symmetric spaces).
result Only HPn and CP2 admit almost quaternionic structures. 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.
Paper finds obstructions to compact Clifford-Klein forms for tangential symmetric spaces.
problem Existence of compact Clifford-Klein forms for tangential symmetric spaces.
method Analyzes tangential homogeneous spaces and provides necessary conditions for their existence.
result New tangential symmetric spaces are found that do not admit compact Clifford-Klein forms.
We prove the existence and uniqueness of geometric models of local isometry classes of locally homogeneous spaces with sectional curvature ∣sec∣≤1. Moreover, we show that the set of geometric models is compact in the pointed C1,α-topology.
A homogeneous space G/H is said to have a compact Clifford-Klein form if there exists a discrete subgroup D of G that acts properly discontinuously on G/H, such that the quotient space D\G/H is compact. When n is even, we find every closed, connected subgroup H of G = SO(2,n), such that G/H has a compact Clifford-Klein…
We show that simply connected Riemannian homogeneous spaces of compact semisimple Lie groups with polar isotropy actions are symmetric, generalizing results of Fabio Podesta and the third named author. Without assuming compactness, we give a classification of Riemannian homogeneous spaces of semisimple Lie groups whose…
Positive simplicial volume implies locally symmetric space structure.
problem Understanding simplicial volume in locally homogeneous spaces.
method Analyzing properties of locally homogeneous Riemannian manifolds.
result Closed locally homogeneous manifolds with positive simplicial volume are locally symmetric.
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 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.
Formula derived for Laplace-Beltrami spectrum on homogeneous spaces.
problem Calculating the spectrum of the Laplace-Beltrami operator on homogeneous spaces.
method Formula derivation based on eigenvalues of a generalized Casimir operator and spherical representations.
result First detailed computation and investigation of the spectrum for a family of metrics on the Aloff-Wallach manifold.
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…
Study on Einstein metrics on specific homogeneous spaces.
problem Existence of invariant Einstein metrics on aligned homogeneous spaces.
method Analysis of G_1xG_2-invariant Einstein metrics on G_1/K x G_2/K for compact Lie groups.
result Existence of Einstein metrics is equivalent to a real root of a quartic polynomial.
We prove that among all compact homogeneous spaces for an effective transitive action of a Lie group whose Levi subgroup has no compact simple factors, the seven-dimensional flat torus is the only one that admits an invariant torsion-free G2(2)-structure.
A Riemannian manifold is called Weyl homogeneous, if its Weyl tensors at any two points are "the same", up to a positive multiple. A Weyl homogeneous manifold is modeled on a homogeneous space M0, if its Weyl tensor at every point is "the same" as the Weyl tensor of M0, up to a positive multiple. We prove that a …
Using a K-theory point of view, Bott related the Atiyah-Singer index theorem for elliptic operators on compact homogeneous spaces to the Weyl character formula. This article explains how to prove the local index theorem for compact homogenous spaces using Lie algebra methods. The method follows in outline the proof of …