arXiv research

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

Local equivalence shown between specific distributions and flat Cartan distribution.

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

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

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 describes conformal structures and Pfaffian systems for rolling surfaces.

Classifies maximal antipodal sets in symmetric spaces.

In 1910 E. Cartan constructed the canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions on a 5-dimensional manifold. We solve the analogous problems for rank 2 distributions on an n-dimensional manifold for arbitrary n greater than 5. Our method is a kind of symplectification…

New $G_2$ symmetry found in 8D distribution with 6D square.

Study maximal antipodal sets in exceptional symmetric spaces.

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

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.

Constructs canonical frames for specific distributions, proving maximality and describing germs.

New algorithm efficiently estimates symmetric properties using approximate PML.

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…

Classifies R-spaces with a specific symmetric structure.

We show that for n>2 the following equivalence problems are essentially the same: the equivalence problem for Lagrangians of order n with one dependent and one independent variable considered up to a contact transformation, a multiplication by a nonzero constant, and modulo divergence; the equivalence problem for the s…

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.

New RESK distributions improve robust clustering of skewed data.

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…

Local normal forms for symmetrical contact structures on 3-manifolds.

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

We demonstrate how the novel approach to the local geometry of structures of nonholonomic nature, originated by Andrei Agrachev, works in the following two situations: rank 2 distributions of maximal class in R^n with non-zero generalized Wilczynski invariants and rank 2 distributions of maximal class in R^n with addit…

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.

In this paper several examples of gaps (lacunes) between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the symmetries of geometric structures and differential equations. A general result c…

New classification of conformal structures with maximal $G_2$ symmetry.

Consider a Riemannian symmetric space $X= G/K$ of non-compact type, where $G$ denotes a connected, real, semi-simple Lie group with finite center, and $K$ a maximal compact subgroup of $G$. Let $\widetilde X$ be its Oshima compactification, and $(π,\mathrm{C}(\widetilde X))$ the regular representation of $G$ on $\widet…

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

Optimal stopping times maximize/minimize Brownian motion distance between radially symmetric marginals.

We describe a hyperbolic version of the Ambartzumian-Pleijel identity. We use this identity to prove the hyperbolic Crofton formula and the hyperbolic isoperimetric inequality. This identity also provides a way to compute the chord length distribution for an ideal polygon in the hyperbolic plane. The analogous results …

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.

Paper introduces AQ-LRMF for better noise modeling in low-rank matrix factorization.

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.