Characterizes invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.
problem Identifying invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.
method Characterization through invariant Riemannian metrics and Killing vector fields.
result A family of invariant contact metric structures is obtained on tangent sphere bundles of compact symmetric spaces with rank greater than or equal to two.
Proves stability of Einstein metrics on specific symmetric spaces.
problem Stability of Einstein metrics on symmetric spaces of compact type.
method Linear stability analysis of the Einstein-Hilbert action.
result Resolves stability problem for irreducible symmetric spaces of compact type.
Compact holonomy groups found in symmetric spaces.
problem Characterizing holonomy groups in symmetric spaces.
method Analyzing the holonomy group structure of locally symmetric spaces.
result Holonomy groups are compact and have finite index in the orthogonal group.
Counting geodesics on compact symmetric spaces using orbit dimensions and topological data.
problem Counting geodesics on compact symmetric spaces.
method Using orbit dimensions and topological data of the symmetric space.
result Obtained data on dimensions and connected components of focal orbits.
Paper constructs an example of a non-compact submanifold in a quaternionic Kähler symmetric space.
problem Tackles the construction of a non-compact totally complex submanifold in a quaternionic Kähler symmetric space.
method Uses an isometric action of a compact Lie group and a maximal totally geodesic sphere.
result Proves the existence of a non-compact totally complex submanifold of maximal dimension in a compact quaternionic Kähler symmetric space.
We prove that a polar foliation of codimension at least three in an irreducible compact symmetric space is hyperpolar, unless the symmetric space has rank one. For reducible symmetric spaces of compact type, we derive decomposition results for polar foliations.
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.
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.
New spectral analysis on non-compact spaces.
problem Analyzing pseudo-Riemannian locally symmetric spaces.
method Initiating spectral analysis beyond classical settings.
result Recent results in non-compact spaces.
Study of a 32D Rosenfeld projective plane, a symmetric space.
problem Understanding the properties of a specific symmetric space.
method Detailed study and analysis of the geometric structure.
result Comprehensive understanding of the 32D Rosenfeld projective plane.
New compact minimal submanifolds found in Riemannian symmetric spaces.
problem Finding compact minimal submanifolds in Riemannian symmetric spaces.
method Constructing multi-dimensional families of compact minimal submanifolds via complex-valued eigenfunctions.
result New families of compact minimal submanifolds of codimension two in SU(n)/SO(n), Sp(n)/U(n), SO(2n)/U(n), and SU(2n)/Sp(n). Paper proves geodesic ball maximizes second Robin eigenvalue in non-compact symmetric spaces.
problem Maximizing the second Robin eigenvalue in non-compact rank-1 symmetric spaces.
method Quantitative spectral inequality for the second Robin eigenvalue.
result Geodesic ball maximizes the second Robin eigenvalue among domains of the same volume.
Study string topology on symmetric spaces, showing non-triviality and nilpotency results.
problem Understanding the structure of string topology on symmetric spaces.
method Used cycles from Bott-Samelson and Ziller to study coproduct and product.
result Showed non-triviality and nilpotency of Chas-Sullivan product and coproduct for higher rank symmetric spaces.
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 give an explicit classification of maximal antipodal sets in any irreducible compact symmetric space except for spin groups and half spin groups, and some quotient symmetric spaces associated to them.
New examples of isoparametric families on non-compact symmetric spaces.
problem Constructing isoparametric families with non-austere focal sets.
method New extension method from Euclidean spaces to symmetric spaces of non-compact type.
result First examples of isoparametric families on non-compact symmetric spaces.
In two previous papers, we started a study of the first eigenvalue of the Dirac operator on compact spin symmetric spaces, providing, for symmetric spaces of "inner" type, a formula giving this first eigenvalue in terms of the algebraic data of the groups involved. We conclude here that study by giving the explicit exp…
Study on deformations of symmetric spaces using Jordan algebras.
problem Deformability of symmetric Einstein metrics on compact Lie algebras.
method Developed sandwich operators and quadratic Casimir operators for compact Lie algebras; calculated obstruction integrals from invariant polynomials; explored relation to simple Jordan algebras.
result Proved the nonlinear instability of most infinitesimally deformable irreducible compact symmetric spaces.
Method calculates spectra of Rarita-Schwinger operator on symmetric spaces.
problem Calculating spectra of the Rarita-Schwinger operator on compact symmetric spaces.
method Using Weitzenböck formulas, Laplace operator, Casimir operator, Freudenthal's formula, and branching rules.
result Obtained spectra on the sphere, complex projective space, and quaternionic projective space.
We prove that an isometric action of a compact Lie group on a compact symmetric space is variationally complete if and only if it is hyperpolar.
Volume comparison theorem for rank 1 symmetric spaces proved.
problem Volume comparison for symmetric spaces of non-compact type.
method Normalized Ricci--DeTurck flow to analyze volume functional and derive monotonicity properties.
result Volume comparison theorem established for rank 1 symmetric spaces of non-compact type.
We study hyperpolar actions on reducible symmetric spaces of the compact type. Our main result is that an indecomposable hyperpolar action on a symmetric space of the compact type is orbit equivalent to a Hermann action or of cohomogeneity one.
In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. This identity is a generalization of Cartan-D'Atri's identity for curvature-adapted(=amenable) isoparametric hypersurfaces in rank one symmetric spaces. Furthermore,…
Compact rank one symmetric spaces are rigid under certain curvature conditions.
problem Rigidity of compact rank one symmetric spaces under curvature constraints.
method Examined compact symmetric spaces with metric g0 of rank one, and another metric g with sectional curvature bounded by 0 to 1. result If g equals g0 outside a convex subset, then g is isometric with g0. New Einstein manifolds split into symmetric and compact parts.
problem Understanding Einstein manifolds with unimodular isometry groups.
method Theory of polar actions, Lie-theoretic arguments, and maximum principles.
result Negative Einstein manifolds split into symmetric and compact parts.
We give a necessary and sufficient condition for orbits of commutative Hermann actions and actions of the direct product of two symmetric subgroups on compact Lie groups to be biharmonic in terms of symmetric triad with multiplicities. By this criterion, we determine all the proper biharmonic submanifolds in irreducibl…
In this paper, we prove that full irreducible curvature-adapted isoparametric submanifolds of codimension greater than one in a symmetric space of non-compact type are principal orbits of Hermann actions on the symmetric spaces under certain conditions.
We prove that the orbits of a polar action of a compact Lie group on a compact rank one symmetric space are tautly embedded with respect to Z_2-coefficients.
Harmonic functions on compact symmetric spaces exhibit strong convexity properties.
problem Understanding the convexity of harmonic functions on compact symmetric spaces.
method Analyzing the nonnegativity of the Laplacian powers of harmonic functions.
result Harmonic functions on compact symmetric spaces have nonnegative Laplacian powers, demonstrating strong convexity.
We obtain the full classification of coisotropic and polar actions of compact Lie group on irreducible Hermitian symmetric spaces.
Classifies 4D spaces with compact Clifford-Klein forms.
problem Classifying 4D symmetric spaces with compact Clifford-Klein forms.
method Developed a method for 1-connected solvable symmetric spaces.
result Classification of 4D spaces with compact Clifford-Klein forms.
Researchers found all invariant contact structures on tangent sphere bundles of compact symmetric spaces.
problem Identifying all invariant contact metric structures on tangent sphere bundles of compact rank-one symmetric spaces.
method Explicitly obtained all structures, distinguishing K-contact, Sasakian, and 3-Sasakian structures.
result There is a unique Sasakian-Einstein metric on tangent sphere bundles of spheres and real projective spaces.
Systematic prolongation for Killing two-tensors in symmetric spaces.
problem Understanding Killing two-tensors in symmetric spaces.
method Systematic prolongation procedure for Killing two-tensors, focusing on locally symmetric spaces.
result Natural quadratic mapping from Killing fields to Killing two-tensors on irreducible locally symmetric spaces of compact type.
Constructs minimal submanifolds in symmetric spaces using eigenfunctions.
problem Finding minimal submanifolds in symmetric spaces.
method Employing recent results from S. Gudmundsson and T.J. Munn, constructing submanifolds using eigenfunctions.
result Constructs minimal submanifolds of classical compact Riemannian symmetric spaces.
We prove that a function on an irreducible compact symmetric space M, which is not a sphere, is determined by its integrals over the shortest closed geodesics in M. We also prove a support theorem for the Funk transform on rank one symmetric spaces which are not spheres.
Near isospectrality forces full isospectrality for compact quotients of symmetric spaces.
problem Inverse spectral problem for Riemannian manifolds
method Proving near isospectrality implies full isospectrality
result Compact quotients of symmetric spaces have full isospectrality
We analyze polar actions on Hermitian and quaternion-Kähler symmetric spaces of compact type. For complex integrable polar actions on Hermitian symmetric spaces of compact type we prove a reduction theorem and several corollaries concerning the geometry of these actions. The results are independent of the classificatio…
Paper proves eigenvalue inequality for Hopf-symmetric domains.
problem Eigenvalue inequality for Hopf-symmetric domains in non-compact symmetric spaces.
method Used geometric and spectral analysis on non-compact rank one symmetric spaces.
result Eigenvalue inequality for bounded Hopf-symmetric domains in non-compact symmetric spaces.
Study proper actions of Lie groups on symmetric spaces, finding rigidity results and Hurwitz-Radon numbers.
problem Proper actions of non-compact semisimple Lie groups on pseudo-Riemannian symmetric spaces.
method Analysis of symmetric spaces and rigidity results.
result Any connected non-compact semisimple Lie group acting properly on these spaces must be globally isomorphic to Spin(n,1) up to compact factors. Develops sublinear Morse theory in symmetric spaces.
problem Understanding sublinear Morse properties in symmetric spaces.
method Theory of sublinearly Morse boundary and lemma in higher rank symmetric spaces.
result Proves sublinear Morse lemma in higher rank symmetric spaces.
Proves gap rigidity theorem for Hermitian symmetric spaces.
problem Gap rigidity problems in compact Hermitian symmetric spaces.
method Dual analogy to Mok's noncompact case theorem, theorem on higher dimensional submanifolds.
result Proves gap rigidity theorem for diagonal curves in tube type spaces.
Cohomogeneity-one actions on symmetric spaces of mixed type
problem Classifying cohomogeneity-one actions on symmetric spaces
method Using a new family of diagonal cohomogeneity-one actions
result Reducing the classification problem to symmetric spaces of a single type
We show that closed, immersed, minimal hypersurfaces in a compact symmetric space satisfy a lower bound on the index plus nullity, which depends linearly on their first Betti number. Moreover, if either the minimal hypersurface satisfies a certain genericity condition, or if the ambient space is a product of two CROSSe…
Studying the isotropy orbits of compact symmetric spaces Reiswich introduced a family of explicit polynomials in one variable in order to describe the unique minimal isotropy orbit of compact symmetric spaces with Dynkin diagram of type Dm. Based on this geometric interpretation he conjectured that these polynomials…
The phase space of a compact, irreducible, simply connected, Riemannian symmetric space admits a natural family of Kähler polarizations parametrized by the upper half plane S. Using this family, geometric quantization, including the half-form correction, produces the field Hcorr→S of quantum Hilbert s…
The isotropy action on certain symmetric spaces is shown to be equivariantly formal.
problem Understanding the equivariant formality of isotropy actions on symmetric spaces.
method Developed a new approach to prove equivariant formality for (Z2⊕Z2)-symmetric spaces. result Symmetric spaces with (Z2⊕Z2)-symmetry are equivariantly formal and formal in the Sullivan sense. We give a formula for the first eigenvalue of the Dirac operator acting on spinor fields of a spin compact irreducible symmetric space G/K.
We prove an equidistribution result for totally geodesic submanifolds in a compact locally symmetric space. In the case of Hermitian locally symmetric spaces, this gives a convergence theorem for currents of integration along totally geodesic subvarieties. As a corollary, we obtain that on a complex surface which is a …