Study the smallest Laplace eigenvalue in special geometric spaces.
problem Finding the smallest positive eigenvalue of Laplace-Beltrami operator in strongly isotropy irreducible spaces.
method Explicit expression for simply connected cases, proving Einstein manifold properties and eigenvalue bounds.
result Proved E<λ1≤16E for all strongly isotropy irreducible 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.
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.
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.
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.
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. 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 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.
Study classifies compact geodesic orbit spaces with two isotropy components.
problem Characterizing geodesic orbit Riemannian spaces.
method Classification of spaces with specific isotropy properties.
result Classification of compact geodesic orbit spaces with two isotropy summands.
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 …
This paper proves rational eigenvalues for shape operators in certain spaces.
problem Understanding rational properties of shape operators in symmetric spaces.
method Analyzing normal holonomy and shape operators in singular orbits of isotropy representations.
result Shape operators have rational eigenvalues in specific normal holonomy factors.
We study the isotropy representation of real flag manifolds associated to simple Lie algebras that are split real forms of complex simple Lie algebras. For each Dynkin diagram the invariant irreducible subspaces for the compact part of the isotropy subgroup are described. Contrary to the complex flag manifolds the deco…
We consider the question: can the isotropy representation of an irreducible pseudo-Riemannian symmetric space be realized as a conformal holonomy group? Using recent results of Cap, Gover and Hammerl, we study the representations of SO(2,1), PSU(2,1) and PSp(2,1) as isotropy groups of irreducible symmetric spaces of si…
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 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…
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.
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 consider a homogeneous fibration G/L→G/K, with symmetric fiber and base, where G is a compact connected semisimple Lie group and L has maximal rank in G. We suppose the base space G/K is isotropy irreducible and the fiber K/L is simply connected. We investigate the existence of G-invariant Einstein…
Extends particle classification to curved space-times using groupoids.
problem Classifying elementary particles in curved space-time.
method Developed a new definition of elementary particles as irreducible projective representations of kinematical groupoids, extending Wigner's program.
result Classification of elementary particles valid for a wide range of space-times, including new massless particles in magnetic-like backgrounds.
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 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.
Classifies Lie group representations linked to quaternion-Kähler symmetric spaces.
problem Classifying representations of Lie groups with specific orbit spaces.
method Analyzing representations of Sp(1)k-extensions and quaternion-Kähler symmetric spaces. result Representations are derived from isotropy representations of quaternion-Kähler symmetric spaces.
The paper classifies flag manifolds with specific isotropy components and finds conditions for Kähler-like scalar curvature.
problem Classifying flag manifolds with specific isotropy components and finding conditions for Kähler-like scalar curvature.
method Investigating invariant almost Hermitian structures on generalized flag manifolds with two or three irreducible components.
result Classification of flag manifolds admitting Kähler-like scalar curvature and conditions for such structures.
Solves Einstein metric problem on specific manifolds.
problem Finding invariant Einstein metrics on cohomogeneity one manifolds.
method Assumes irreducible isotropy representations, uses G-invariance. result Shows existence of G-invariant Einstein metrics. We prove a Berger type theorem for the normal holonomy group (i.e., the holonomy group of the normal connection) of a full complete complex submanifold of the complex projective space. Namely, if the normal holonomy does not act transitively, then the submanifold is the complex orbit, in the complex projective space, o…
We are studying a relationship between isoparametric hypersurfaces in spheres with four distinct principal curvatures and the moment maps of certain Hamiltonian actions. In this paper, we consider the isoparametric hypersurfaces obtained from the isotropy representations of compact irreducible Hermitian symmetric space…
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.
In this paper, we show that small spherical soap bubbles in irreducible simply connected symmetric spaces of rank greater than one are constructed from the limits of a certain kind of modified mean curvature flows starting from small spheres in the Euclidean space of dimension equal to the rank of the symmetric space, …
Let G be a simple compact connected Lie group. We study homogeneous Einstein metrics for a class of compact homogeneous spaces, namely generalized flag manifolds G/H with second Betti number b2(G/H)=1. There are 8 infinite families G/H corresponding to a classical simple Lie group G and 25 exceptional flag…
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.
Study on G2 actions on symmetric spaces, focusing on orbit properties.
problem Investigating properties of orbits in symmetric spaces related to G2. method Classification and analysis of orbits as Riemannian submanifolds, focusing on principal curvatures and specific types of orbits.
result Classification and properties of orbits in symmetric spaces related to G2. New conditions for Ricci curvature on homogeneous spaces.
problem Existence of metrics with prescribed Ricci curvature.
method Sufficient and necessary conditions for existence of metrics with prescribed Ricci curvature.
result Conditions for existence of metrics with prescribed Ricci curvature.
The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using a normal holonomy approach. Indeed, we explain how these submanifolds can be reg…
We give an overview of our earlier classification results in [DW4] and [DW6] for superpotentials of scalar curvature type of the cohomogeneity one Ricci-flat equations. We then give an account of the classification in the case where the isotropy representation of the principal orbit consists of exactly three distinct i…
For a compact connected Lie group G we study the class of bi-invariant affine connections whose geodesics through e∈G are the 1-parameter subgroups. We show that the bi-invariant affine connections which induce derivations on the corresponding Lie algebra g coincide with the bi-invariant metric connecti…
There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical bui…
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. Study Einstein metrics on HimesH/ΔK spaces.
problem Existence and classification of invariant Einstein metrics on HimesH/ΔK. method Investigate HimesH-invariant Einstein metrics on M=HimesH/ΔK. result Find unstable Einstein metrics on M for many spaces H/K. The study confirms a conjecture about polynomials related to symmetric spaces.
problem Understanding polynomials associated with isotropy orbits of symmetric spaces.
method Identified Reiswich's polynomials as special cases of Jacobi polynomials and proved their conjecture.
result The polynomials have pairwise different real roots in the interval [0,1].
Eigenvalue problem for Kähler metrics on compact manifolds.
problem Eigenvalue problem for the Laplacian on Kähler manifolds.
method Introducing λk-extremal Kähler metrics and deducing conditions for extremality. result Conditions for a Kähler metric to be λk-extremal. The paper classifies submanifolds in symmetric spaces without analyticity.
problem Classifying submanifolds in symmetric spaces of non-compact type.
method Building theory and analysis of reflective focal submanifolds.
result Submanifolds are principal orbits of Hermann type actions.
Estimates average number of common zeros of Laplacian eigenfunctions on manifolds.
problem Estimating the average number of common zeros of Laplacian eigenfunctions on compact Riemannian manifolds.
method Application of Crofton's formula for the sphere.
result Proves an estimate for the average number of common zeros of eigenfunctions, showing it does not exceed a specific expression.
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…
The classification of Riemannian manifolds by the holonomy group of their Levi-Civita connection picks out many interesting classes of structures, several of which are solutions to the Einstein equations. The classification has two parts. The first consists of isolated examples: the Riemannian symmetric spaces. The sec…
The paper finds conditions for biharmonic orbits in symmetric spaces.
problem Conditions for biharmonic orbits in symmetric spaces.
method Analyzes isotropy representations and biharmonic submanifolds in hyperspheres.
result Necessary and sufficient conditions for biharmonic orbits in symmetric 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.