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.
Classified spaces in low dimensions.
problem Irreducible homogeneous almost Hermite-Lorentz spaces in low dimensions.
method Classification through complex dimension 3.
result Classification of spaces in low dimensions.
New proof shows no negative curvature Einstein metrics in specific dimensions.
problem Proving nonexistence of certain Einstein metrics in 9 and 10 dimensions.
method Cohomogeneity-one approach to show nonexistence of negative curvature Einstein metrics.
result Noncompact homogeneous spaces not diffeomorphic to Euclidean space of dimension 9 or 10 admit no homogeneous Einstein metrics of negative Ricci curvature, with only three potential exceptions.
Study on CR structures in 7D, proving maximal symmetry dimension.
problem Proving symmetry dimension bound for CR structures in 7D.
method Investigating homogeneous models and proving uniqueness.
result 8 is the maximal symmetry dimension of 3-nondegenerate CR structures in 7D.
Study finds all hyperbolic manifolds with specific automorphism group dimensions.
problem Classifying homogeneous Kobayashi-hyperbolic manifolds based on automorphism group size.
method Analyzing holomorphic automorphism groups of dimension n2−3 for manifolds of dimension n≥2. result All connected homogeneous Kobayashi-hyperbolic manifolds of dimension n≥2 with automorphism group of dimension n2−3 are identified. Study finds all specific hyperbolic manifolds with high automorphism groups.
problem Classifying hyperbolic manifolds with high-dimensional automorphism groups.
method Examined homogeneous Kobayashi-hyperbolic manifolds of dimension n≥2. result Determined all such manifolds with holomorphic automorphism group dimension n2−2. In this article we classify expanding homogeneous Ricci solitons up to dimension 5, according to their presentation as homogeneous spaces. We obtain that they are all isometric to solvsolitons, and this in particular implies that the generalized Alekseevskii conjecture holds in these dimensions. In addition, we prove t…
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.
Researchers found all special metrics in 4D for certain curvature functionals.
problem Identifying special metrics in 4D for quadratic curvature functionals.
method Determined all homogeneous metrics that are critical for quadratic curvature functionals.
result All homogeneous metrics in 4D for some quadratic curvature functionals have been identified.
No non-product Hessian rank 1 affine homogeneous hypersurfaces exist in dimensions 5 and above.
problem Identifying non-product Hessian rank 1 affine homogeneous hypersurfaces in higher dimensions.
method Developed a normal form for hypersurfaces under the affine group, up to order ≤ n+5, in any dimension n ≥ 2.
result Non-existence of non-product Hessian rank 1 affine homogeneous hypersurfaces in dimensions 5 and above.
Expanding on subgroup dimension, this paper covers new dimensions and compactness.
problem Understanding subgroup dimensions and their properties.
method Generalization of dimension datum, study of disconnected subgroups, and isospectral sets.
result New insights into subgroup dimensions and compactness of isospectral sets.
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. …
We prove that there is no nontrivial homogeneous order 2 solutions of fully nonlinear uniformly elliptic equations in dimension 4.
The study classifies Hessian rank 1 hypersurfaces in dimensions 2, 3, and 4.
problem Classifying Hessian rank 1 affinely homogeneous hypersurfaces in specific dimensions.
method Power Series Method of Equivalence, infinitesimal calculations.
result Identified all non-product constant Hessian rank 1 affinely homogeneous hypersurfaces in dimensions 2, 3, and 4.
A method, due to Élie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2,2) signature are Einstein of which one is Ricci-flat. If…
Spinorially constructs Sasakian and 3-Sasakian structures in arbitrary dimensions.
problem Constructing Sasakian and 3-Sasakian structures in arbitrary dimensions.
method Spinorial construction and analysis of invariant spinors.
result Complete description of invariant spinors on homogeneous 3-Sasakian spaces.
Study local properties of homogeneous ANR-spaces, proving dimension full-valuedness.
problem Understand local structure of homogeneous ANR-spaces.
method Describe local structure and use it to prove dimension full-valuedness.
result Every finite-dimensional homogeneous metric ANR-compactum is dimensionally full-valued.
Several large classes of homogeneous spaces are known to be formal---in the sense of Rational Homotopy Theory. However, it seems that far fewer examples of non-formal homogeneous spaces are known. In this article we provide several construction principles and characterisations for non-formal homogeneous spaces, which w…
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…
Generalizes Alexandroff's Vn-continua to cohomological dimensions.
problem Extending Alexandroff's concept of Vn-continua to cohomological dimensions. method Proves that strongly locally homogeneous generalized continua with cohomological dimension n are generalized Vn-spaces. result Every strongly locally homogeneous continuum of covering dimension n is a Vn-continuum in the sense of Alexandroff. Researchers find solutions to Einstein equations in higher dimensions.
problem Finding spatially homogeneous solutions to vacuum Einstein equations in general dimensions.
method Assumed spatially homogeneous spacetime, solved Einstein equations for globally hyperbolic spacetimes with specific symmetry groups.
result Spatially homogeneous solutions found, corresponding to Bianchi type II in 4D, and constraints on spacetime expansion.
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…
Let G be a compact Lie group acting isometrically on a compact Riemannian manifold M with nonempty fixed point set MG. We say that M is fixed-point homogeneous if G acts transitively on a normal sphere to some component of MG. Fixed-point homogeneous manifolds with positive sectional curvature have been c…
Classifies homogeneous CR hypersurfaces in low dimensions with maximal symmetry.
problem Classifying CR hypersurfaces with maximal symmetry in low dimensions.
method Introduced modified CR symbols to organize local invariants, classified hypersurfaces through modified symbols, and used Lie group structures.
result Found nine model structures among locally homogeneous 2-nondegenerate hypersurfaces in C4. It is proved that no region of a homogeneous locally compact, locally connected metric space can be cut by an Fσ-subset of a "smaller" dimension. The result applies to different finite or infinite topological dimensions of metrizable spaces.
Study geometric vector fields on 5D 2-step nilmanifolds.
problem Analyzing geometric vector fields on specific nilmanifolds.
method Examine vector fields on 2-step nilmanifolds of dimension 5.
result Characterize geometric vector fields on 5D 2-step nilmanifolds.
The dynamics defined by a force field which is positively homogeneous of degree −3 can always be reduced, by simply constraining it. The dimension of the phase space is reduced by two dimensions, while it may only be reduced by one dimension if the degree of homogeneity is different from −3. This remark is an elega…
The paper identifies curvature patterns in 3D homogeneous spaces.
problem Determining curvature forms in 3D locally homogeneous Lorentzian manifolds.
method Analyzing the sectional curvature operator on these manifolds.
result Identified admissible forms of the sectional curvature operator.
We prove Gray & Wolf's conjecture that a Riemannian homogeneous manifold admitting a strict nearly Kahler structure is 3-symmetric. We actually classify them in dimension 6 and use previous results of Swann, Cleyton and Nagy to prove the conjecture in higher dimensions.
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.
Study finds maximal symmetry groups for CR structures with specific properties.
problem Determining the maximal dimension of symmetry groups for CR structures.
method Proved the sharp upper bound for the dimension of symmetry groups for homogeneous, 2-nondegenerate CR manifolds.
result The maximal dimension is n2+7 for n≥3. New compact ECS manifolds with rank 2 discovered, differing from previous rank 1 examples.
problem Finding new compact ECS manifolds with rank 2.
method Constructing new examples of compact pseudo-Riemannian manifolds with parallel Weyl tensor, rank 1 or 2.
result New compact ECS manifolds of rank 2, locally homogeneous, and geodesically incomplete.
We show that pseudo-Riemannian almost quaternionic homogeneous spaces with index 4 and an H-irreducible isotropy group are locally isometric to a pseudo-Riemannian quaternionic Kähler symmetric space if the dimension is at least 16. In dimension 12 we give a non-symmetric example.
The study classifies homogeneous Sasaki manifolds over quaternionic Kähler spaces.
problem Classifying homogeneous Sasaki manifolds over quaternionic Kähler spaces.
method Locally defined Riemannian submersions and homogeneous space constructions.
result Complete classification of homogeneous Sasaki manifolds in the non-degenerate case.
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.
Classifies homogeneous Levi non-degenerate hypersurfaces in complex 3-space.
problem Classifying specific types of complex hypersurfaces.
method Analyzing hypersurfaces with symmetry algebra of dimension at least 6.
result All such hypersurfaces are classified.
We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show the soliton is rigid in dimensions three and four. In the steady case, we give a complete classification in dimension three.
We prove that any homogeneous order one solution to 3-d nondivergence elliptic equations must be linear.
Study on harmonic maps from surfaces to homogeneous spaces, focusing on bubble formation and geometric constraints.
problem Understanding the behavior of harmonic maps from surfaces to homogeneous spaces, especially in the presence of bubbles.
method Refined asymptotic expansions and obstruction relations for sequences developing a single bubble, geometric constraints for weakly conformal maps.
result New geometric constraints on the tangent planes of the limit map and bubble, depending on the dimensionality.
New findings on plane waves in 3D spacetimes, showing non-unimodular elliptic plane waves are unique.
problem Classifying Lorentz homogeneous spaces of dimension 3, focusing on plane waves.
method Revisiting and relaxing usual completeness assumptions, characterizing homogeneous plane waves.
result Non-unimodular elliptic plane waves are unique and non-extendable, geodesically complete only if symmetric.
Researchers classify curvature homogeneous metrics on 4D manifolds.
problem Classifying curvature homogeneous metrics on 4D manifolds.
method Using equivariant diffeomorphisms and cohomogeneity one actions.
result Found that metrics are either symmetric or a specific example by Tsukada.
We prove that any holomorphic locally homogeneous geometric structure on a complex torus, modelled on a complex homogeneous surface, is translation invariant. We conjecture that this result is true is any dimension. In higher dimension we prove it here for nilpotent models. We also prove that in any dimension the trans…
In a recent paper, it was claimed that any homogeneous Finsler space of odd dimension admits a homogeneous geodesic through any point. For the proof, the algebraic method dealing with the reductive decomposition of the Lie algebra of the isometry group was used. However, the proof contains a serious gap. In the present…
Classifies hyperbolic manifolds with specific automorphism groups.
problem Classifying Kobayashi-hyperbolic manifolds with high-dimensional automorphism groups.
method Analyzes manifolds of dimension n≥2 with automorphism groups of dimensions n2−7 or n2−8. result Completes the classification for automorphism groups n2−7 and n2−8. Homogenized SGD explains SGD dynamics in high dimensions.
problem Understanding SGD dynamics in high-dimensional settings.
method Developed a homogenized SGD model to analyze high-dimensional SGD.
result Convergent high-dimensional SGD to homogenized SGD for quadratic statistics.
We describe the holonomy algebras of all canonical connections of homogeneous structures on real hyperbolic spaces in all dimensions. The structural results obtained then lead to a determination of the types, in the sense of Tricerri and Vanhecke, of the corresponding homogeneous tensors. We use our analysis to show th…
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.
New Galilean spacetimes found as pp-wave reductions.
problem Understanding isotropic homogeneous Galilean spacetimes.
method Null reductions of pp-wave spacetimes.
result Found novel torsional Galilean spacetimes.