arXiv research

Characterizes invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.

Proves stability of Einstein metrics on specific symmetric spaces.

Compact holonomy groups found in symmetric spaces.

Counting geodesics on compact symmetric spaces using orbit dimensions and topological data.

Paper constructs an example of a non-compact submanifold in a 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.

New tools compute index of symmetry in homogeneous fibrations.

Study of a 32D Rosenfeld projective plane, a symmetric space.

New compact minimal submanifolds found in Riemannian symmetric spaces.

Paper proves geodesic ball maximizes second Robin eigenvalue in non-compact symmetric spaces.

Study string topology on symmetric spaces, showing non-triviality and nilpotency results.

Paper finds obstructions to compact Clifford-Klein forms for tangential symmetric spaces.

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.

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.

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.

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.

New 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.

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.

Researchers found all invariant contact structures on tangent sphere bundles of compact symmetric spaces.

Systematic prolongation for Killing two-tensors in symmetric spaces.

Constructs minimal submanifolds in symmetric spaces using eigenfunctions.

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.

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.

Study proper actions of Lie groups on symmetric spaces, finding rigidity results and Hurwitz-Radon numbers.

Develops sublinear Morse theory in symmetric spaces.

Proves gap rigidity theorem for Hermitian symmetric spaces.

Cohomogeneity-one actions on symmetric spaces of mixed 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 $D_m$. 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 $H^{corr}\rightarrow S$ of quantum Hilbert s…

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 …

In this note, we show that for any harmonic map into a non-compact symmetric space one can find naturally a "dual" harmonic map into a compact symmetric space which can be constructed from the same basic data (called "potentials" in the loop group formalism). Locally also the inverse/converse duality theorem holds.

I show that any complex manifold that resembles a rank two compact Hermitian symmetric space (other than a quadric hypersurface) to order two at a general point must be an open subset of such a space.

In this paper, we develop a loop group description of harmonic maps $\mathcal{F}: M \rightarrow G/K$ ``of finite uniton type", from a Riemann surface $M$ into inner symmetric spaces of compact or non-compact type. This develops work of Uhlenbeck, Segal, and Burstall-Guest to non-compact inner symmetric spaces. To be mo…