Symplectic resolves orbifolds with uniform isotropy.
problem Symplectic resolution of orbifolds with homogeneous isotropy.
method Constructing symplectic resolutions for orbifolds with specific isotropy properties.
result Symplectic resolutions achieved for orbifolds with uniform isotropy.
The study classifies Riemannian homogeneous spaces with polar isotropy actions.
problem Characterizing Riemannian homogeneous spaces with polar isotropy actions.
method Analyzing simply connected Riemannian homogeneous spaces of compact semisimple Lie groups and various non-compact spaces.
result Classification and non-polar isotropy actions for specific spaces.
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.
Describes metrics on homogeneous spaces with equivalent isotropy summands.
problem Finding G-invariant metrics on homogeneous spaces with equivalent isotropy summands. method One-to-one correspondence between invariant metrics and inner products on tangent spaces, considering isotropy representations.
result Provides a systematic description of such metrics, simplifying the problem of finding G-invariant Einstein metrics. Classifies 5D homogeneous geometries with specific isotropy properties.
problem Classifying 5D homogeneous geometries with certain isotropy properties.
method Thurston's classification of homogeneous geometries, focusing on irreducible and trivial isotropy representations.
result Identifies 5D geometries with irreducible isotropy as irreducible Riemannian symmetric spaces and those with trivial isotropy as specific solvable Lie groups.
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.
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.
Classifies flat pseudo-Riemannian spaces with specific structures.
problem Classifying homogeneous pseudo-Riemannian spaces with invariant structures.
method Classification based on invariant almost hyper-Hermitian structures and H-irreducible isotropy groups.
result All classified spaces are flat except in dimension 12.
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.
New spaces identified with specific properties.
problem Characterizing homogeneous spaces with quaternionic structures.
method Analyzing pseudo-Riemannian almost quaternionic homogeneous spaces with irreducible isotropy.
result Spaces are locally isometric to quaternionic Kähler symmetric spaces under certain conditions.
We classify invariant almost complex structures on homogeneous manifolds of dimension 6 with semi-simple isotropy. Those with non-degenerate Nijenhuis tensor have the automorphism group of dimension either 14 or 9. An invariant almost complex structure with semi-simple isotropy is necessarily either of specified 6 homo…
Ricci flow on certain homogeneous spaces creates metrics with positive curvature.
problem Finding metrics with positive Ricci curvature on specific homogeneous spaces.
method Normalized Ricci flow on simply connected homogeneous spaces with two equivalent isotropy summands.
result Every G-invariant metric evolves to one with positive Ricci curvature under Ricci flow.
We consider numerical integrators of ODEs on homogeneous spaces (spheres, affine spaces, hyperbolic spaces). Homogeneous spaces are equipped with a built-in symmetry. A numerical integrator respects this symmetry if it is equivariant. One obtains homogeneous space integrators by combining a Lie group integrator with an…
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.
Novel Ricci flow normalization for homogeneous spaces, focusing on flag manifolds.
problem Understanding the limiting behavior and symmetry properties of Ricci flow on homogeneous spaces.
method Introducing a novel normalization for the homogeneous Ricci flow and characterizing Gromov-Hausdorff limits.
result Full classification of Gromov-Hausdorff limits and detailed phase portraits for three-isotropy-summands flag manifolds.
The paper classifies invariant connections and Einstein structures on isotropy irreducible spaces.
problem Classifying invariant connections and Einstein structures on isotropy irreducible spaces.
method Systematic study and classification of invariant affine or metric connections on naturally reductive spaces.
result Classification of invariant metric connections with skew-torsion and abla-Einstein structures. We show that the isotropy action of a homogeneous space G/K, where G and K are compact, connected Lie groups and K is defined by an automorphism on G, is equivariantly formal and that (G,K) is a Cartan pair.
Classifies 5D homogeneous geometries with nontrivial reducible linear isotropy.
problem Classifying 5D homogeneous geometries with specific properties.
method Thorough classification using Thurston's criteria and analysis of linear isotropy representations.
result Found a countably infinite family of geometries diffeomorphic to S3imesS2. We consider the Ricci flow equation for invariant metrics on compact and connected homogeneous spaces whose isotropy representation decomposes into two irreducible inequivalent summands. By studying the corresponding dynamical system, we completely describe the behaviour of the homogeneous Ricci flow on this kind of sp…
Study on irreducibility of Laplacian eigenspaces in homogeneous spaces.
problem Existence of G-invariant Riemannian metrics with irreducible Laplacian eigenspaces. method Analysis of compact homogeneous spaces G/K and their metrics. result Normal metric of rank one symmetric spaces is the only one with irreducible Laplacian eigenspaces.
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.
Study on invariant Einstein metrics on specific flag manifolds.
problem Existence of invariant Einstein metrics on real flag manifolds.
method Analysis of isotropy representations and Riemannian metrics.
result Existence of non-diagonal Einstein metrics on real flag manifolds.
Simplified conditions for GO metrics in homogeneous manifolds.
problem Determine G-GO metrics in compact homogeneous manifolds.
method Simplified conditions for GO metrics based on equivalent isotropy submodules.
result Algebraic conditions for GO metrics in homogeneous manifolds.
In this paper we study the global behavior of the Ricci flow equation for two classes of homogeneous manifolds with two isotropy summands. Using methods of the qualitative theory of differential equations, we present the global phase portrait of such systems and derive some geometrical consequences on the structure of …
Classifies special geometric distributions.
problem Classifying homogeneous real (2,3,5) distributions.
method Multiply transitive classification up to local diffeomorphism.
result Classified multiply transitive homogeneous real (2,3,5) distributions.
New derivation shows spacetime interval is quadratic without light.
problem Deriving spacetime geometry from fundamental principles.
method Formalizing axioms of smoothness, homogeneity, isotropy, and determinism.
result Invariant spacetime interval is quadratic, independent of light.
Study invariant spin^r structures on homogeneous spaces.
problem Classify invariant spin^r structures on homogeneous spaces.
method Introduce and study G-invariance of spin^r structures on homogeneous spaces. result Classification of invariant spin^r structures in terms of isotropy representation.
Study solves Ricci curvature problem for specific noncompact spaces.
problem Solving the Prescribed Ricci Curvature problem for noncompact spaces with two isotropy summands.
method Classified and solved for all simply connected, noncompact G/H with semi-simple G and connected H having two irreducible summands. result Provided solutions to the Prescribed Ricci Curvature problem for all such spaces.
We classify the non-degenerate homogeneous hypersurfaces in real and complex affine four-space whose symmetry group is at least four-dimensional.
Develops a foundational argument for Lorentzian or Euclidean spacetime geometry without light or electromagnetic phenomena.
problem Relativity without light
method Formalizing physical principles as axioms about an invariant interval function D result Invariant interval functions are powers of nondegenerate quadratic forms
The paper finds multiple Einstein metrics on Stiefel manifolds.
problem Invariant Einstein metrics on Stiefel manifolds with specific isotropy summands.
method Viewed as a total space over generalized flag manifolds, the paper proves the existence of at least four invariant Einstein metrics.
result At least four invariant Einstein metrics are found, including two new ones.
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.
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.
We consider four-dimensional homogeneous pseudo-Riemannian manifolds with non-trivial isotropy and completely classify the cases giving rise to non-trivial homogeneous Ricci solitons. In particular, we show the existence of non-compact homogeneous (and also invariant) pseudo-Riemannian Ricci solitons which are not isom…
We study the quasi-convergence equivalence of some families of metrics on locally homogeneous closed 4-manifolds with trivial isotropy group, and identify the dimension of each equivalence class under certain conditions.
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.
Study investigates Einstein flow stability and convergence with matter sources.
problem Stability and convergence of Einstein flow with matter sources.
method Incorporates matter sources into the Einstein flow and examines stability and convergence.
result Similar conclusions can be drawn about the evolution of manifolds to approximate homogeneity and isotropy.
In this paper, we classify compact simply connected cohomogeneity one manifolds up to equivariant diffeomorphism whose isotropy representation by the connected component of the principal isotropy subgroup has three or less irreducible summands. The manifold is either a bundle over a homogeneous space or an irreducible …
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…
A generalized flag manifold is a homogeneous space of the form G/K, where K is the centralizer of a torus in a compact connected semisimple Lie group G. We classify all flag manifolds with four isotropy summands and we study their geometry. We present new G-invariant Einstein metrics by solving explicity the Ei…
Einstein metrics on homogeneous torus bundles
problem Einstein metrics on total space of homogeneous torus bundles
method Descend to common base and establish estimates
result Prove precompactness theorem for Einstein manifolds
We give a positive answer to the Chavel's conjecture [J. Diff. Geom. 4 (1970), 13-20]: a simply connected rank one normal homogeneous space is symmetric if any pair of conjugate points are isotropic. It implies that all simply connected rank one normal homogeneous space with the property that the isotropy action is var…
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…
Classifies tube domains with specific properties in complex spaces.
problem Classifying tube domains with unique geometric properties.
method Analyzes tube domains in complex spaces with homogeneous boundaries and specific Levi forms.
result Classifies tube domains with large automorphism groups in arbitrary dimensions.
We study in this paper three natural notions of convergence of homogeneous manifolds, namely infinitesimal, local and pointed, and their relationship with a fourth one, which only takes into account the underlying algebraic structure of the homogeneous manifold and is indeed much more tractable. Along the way, we intro…
Study homogeneous geodesics in M-spaces, proving standard metric is unique.
problem Investigate homogeneous geodesics in M-spaces. method Analyze properties of isotropy representation and tangent space decomposition.
result For various classes of M-spaces, only standard metric is a geodesic metric. We prove that M. Kramer's classification of list of spherical pairs coincides with that for weakly symmetric spaces by examining the linear isotropy representation of the corresponding homogeneous space associated to each pair.
We construct the homogeneous Einstein equation for generalized flag manifolds G/K of a compact simple Lie group G whose isotropy representation decomposes into five inequivalent irreducible $\Ad(K)$-submodules. To this end we apply a new technique which is based on a fibration of a flag manifold over another flag m…