arXiv research

Classifies maximal antipodal sets in symmetric spaces.

Study maximal antipodal sets in exceptional symmetric spaces.

Proves conjectures about maximal antipodal sets in symmetric and generalised symmetric spaces.

Local equivalence found between maximally symmetric rolling and flat Cartan distributions.

Extends flat submanifold properties from hyperbolic plane to symmetric spaces.

The paper studies billiards in symmetric tables and finds a measure bound for maximizing orbits.

Researchers found a maximal antipodal set of three elements in a 7x7 sphere space.

Local equivalence shown between specific distributions and flat Cartan distribution.

Study maximally symmetric distribution of An-Nurowski surface rolling on a plane.

The study classifies and characterizes totally symmetric sets in the general linear group.

Maximal representations in infinite dimensional Hermitian spaces are studied with boundary maps.

Solves equivalence problem for CR geometries with simple models.

Let X be a symmetric space of non-compact type or a locally finite, strongly transitive Euclidean building, and let B denote the geodesic boundary of X. We reduce the study of visual limits of maximal flats in X to the study of limits of apartments in the spherical building B: this defines a natural, geometric compacti…

We study the canonical complexifications of non-compact Riemannian symmetric spaces G/K by the Grauert tube construction. We determine the maximal such complexification, a domain already constructed in another context by Akhiezer and Gindikin (Math. Ann., 1990), and show that this domain is Stein. We show there is an a…

New solutions to Chazy equations lead to Ricci-flat metrics.

Classifies R-spaces with a specific symmetric structure.

The study finds that certain curved manifolds can be mapped to symmetric spaces.

Study fibrations of projective spaces for maximal representations.

Paper constructs an example of a non-compact submanifold in a quaternionic Kähler symmetric space.

Characterizes circles in self-dual symmetric R-spaces using geometric properties.

Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a totally geodesic submanifold of M. In previous work the authors proved that i(M) is bounded from below by the rank rk(M) of M. In this paper we classify all irreducible Riemannian symmetric spaces M for which the equ…

The study examines the independence of GKM manifolds and symmetric spaces.

All known examples of homogeneous Einstein metrics of negative Ricci curvature can be realized as left-invariant Riemannian metrics on solvable Lie groups. After defining a notion of maximal symmetry among left-invariant Riemannian metrics on a Lie group, we prove that any left-invariant Einstein metric of negative Ric…

Let $M$ be a Riemannian globally symmetric space of compact type, $M'$ its set of maximal flat totally geodesic tori, and $\mathrm{ad}(M)$ its adjoint space. We show that the kernel of the maximal flat Radon transform $τ:L^2(M) \rightarrow L^2(M')$ is precisely the orthogonal complement of the image of the pullback map…

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

Study of minimal surfaces in a specific symmetric space with polynomial growth.

We construct canonical frames and find all maximally symmetric models for a natural generic class of corank 2 distributions on manifolds of odd dimension greater or equal to 7. This class of distributions is characterized by the following two conditions: the pencil of 2-forms associated with the corresponding Pfaffian …

The paper classifies surfaces in a 3-torus with maximal symmetry.

We use maximal periodic flats to show that on a finite volume irreducible locally symmetric manifold of dimension $\geq 3$, no metric $g$ has more symmetry than the locally symmetric metric. We also show that if $g$ is a finite volume metric that is not locally symmetric, then its lift to the universal cover has discre…

We show that any element of the universal Teichmüller space is realized by a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in $AdS^{n+1}$, any subset $E$ of the boundary at infinity which is the boun…

In this paper, we determine the partial positivity(resp., negativity) of the curvature of all irreducible Riemannian symmetric spaces. From the classifications of abstract root systems and maximal subsystems, we can give the calculations for symmetric spaces both in classical types and in exceptional types.

The integrability conditions for the existence of Killing-Yano tensors or, equivalently, covariantly closed conformal Killing-Yano tensors, in the presence of torsion are worked out. As an application, all metrics and torsions compatible with the existence of a Killing-Yano tensor of order n-1 are obtained. Finally, th…

We prove that every non-positively curved locally symmetric manifold M of finite volume contains a compact set K such that no periodic maximal flat can be homotoped out of K.

The Riemannian symmetric space SU_{2,m}/S(U_2U_m) is both Hermitian symmetric and quaternionic Kahler symmetric. Let M be a hypersurface in SU_{2,m}/S(U_2U_m) and denote by TM its tangent bundle. The complex structure of SU_{2,m}/S(U_2U_m) determines a maximal complex subbundle C of TM, and the quaternionic structure o…

The paper describes conformal structures and Pfaffian systems for rolling surfaces.

Symmetric losses improve classifier robustness from corrupted labels.

Classifies totally geodesic submanifolds in exceptional symmetric spaces.

Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on …

For suitable metrics on the locally symmetric space associated to a maximal representation, we prove inequalities between the length of the boundary and the lengths of orthogeodesics that generalize the classical Basmajian's identity from Teichmueller theory. Any equality characterizes diagonal embeddings.

The paper explores symmetric losses for better learning from corrupted labels.

We study the contact geometry of scalar second order hyperbolic equations in the plane of generic type. Following a derivation of parametrized contact-invariants to distinguish Monge-Ampere (class 6-6), Goursat (class 6-7) and generic (class 7-7) hyperbolic equations, we use Cartan's equivalence method to study the gen…

Estimates spacelike surfaces' curvature in de Sitter space.

This thesis studies the symplectic structure of holomorphic coadjoint orbits, and their projections. A holomorphic coadjoint orbit O is an elliptic coadjoint orbit which is endowed with a natural invariant Kählerian structure. These coadjoint orbits are defined for a real semi-simple connected non-compact Lie group G w…

The study proves a geometric result related to Harish-Chandra's theorem.

We construct compactifications of Riemannian locally symmetric spaces arising as quotients by Anosov representations. These compactifications are modeled on generalized Satake compactifications and, in certain cases, on maximal Satake compactifications. We deduce that these Riemannian locally symmetric spaces are topol…

Let M be a compact irreducible Hermitian symmetric space and write M=G/K, with G the group of holomorphic isometries of M and K the stability group of the point of 0 in M. We determine the maximal dimension of a complex projective space embedded in M as a totally geodesic submanifold.

Holomorphic discs converge to maximal surfaces under specific flows.

The X-ray transform on a compact symmetric space M is here inverted by means of an explicit inversion formula. The proof uses the conjugacy of the minimal closed geodesics in M and of the maximally curved totally geodesic spheres in M, proved in Math. Ann. 165 (1966), 309--317.