arXiv research

The paper shows measures equidistribute on affine submanifolds with a rate.

Study the boundary of a submanifold of translation surfaces.

We uncover the lowest order differential invariants of Lagrangian submanifolds under affine symplectic maps, and find out what happens when they are constant.

The paper calculates area Siegel--Veech constants for specific submanifolds of REL zero.

We prove that if the Lyapunov spectrum of the Kontsevich-Zorich cocycle over an affine SL$(2,\mathbb{R})$-invariant submanifold is completely degenerate, i.e. $λ_2 = \cdots = λ_g = 0$, then the submanifold must be an arithmetic Teichmueller curve in the moduli space of Abelian differentials over surfaces of genus three…

The paper explores totally geodesic submanifolds in SPD matrices and their properties.

Researchers discover all affinely homogeneous models for surfaces in 4D space.

Study GL(2,R) orbit closures in hyperelliptic components of abelian differentials.

Study on stabilities and moduli spaces of special affine Legendrian submanifolds.

The paper constructs submanifolds with corners in Delzant polytopes from affine subspaces.

The paper studies extremal hypersurfaces in ellipsoids using centro-affine geometry.

A (flat) affine $3$-manifold is a $3$-manifold with an atlas of charts to an affine space $\mathbb{R}^3$ with transition maps in the affine transformation group $\mathrm{Aff}(\mathbb{R}^3)$. We will show that a connected closed affine $3$-manifold is either an affine Hopf $3$-manifold or decomposes canonically to conca…

In (equi-)affine differential geometry, the most important algebraic invariants are the affine (Blaschke) metric h, the affine shape operator S and the difference tensor K. A hypersurface is said to admit a pointwise symmetry if at every point there exists a linear transformation preserving the affine metric, the affin…

Special Lagrangian submanifolds emerge from K3 surface collapse.

The paper generalizes properties of submanifolds to higher dimensions using affine k-spaces.

The study explores special submanifolds in the probability simplex with unique geometric properties.

We prove some ergodic-theoretic rigidity properties of the action of SL(2,R) on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2,R) is supported on an invariant affine submanifold. The main theorems are inspired by the results of several a…

For a non-uniform lattice in SL(2,R), we consider excursions in cusp neighborhoods of a random geodesic on the corresponding finite area hyperbolic surface or orbifold. We prove a strong law for a certain partial sum involving these excursions. This generalizes a theorem of Diamond and Vaaler for continued fractions. I…

This paper studies singular improper affine spheres from Lagrangian submanifolds, classifying stable singularities.

Abstract mathematical formulas for statistical structures and curvatures.

The paper studies the affine focal set of submanifolds in hypersurfaces, describing conditions for regularity and singularities.

We prove some estimates of the volumes of the sets of translation surfaces of unit area having several independent small saddle connections in a rank one affine submanifold.

Develops algorithms for constructing moving frames and differential invariants for Lie group actions.

The paper solves Bernstein problems for specific submanifolds in high-dimensional spaces.

An (flat) affine $3$-manifold is a $3$-manifold with an atlas of charts to an affine space ${\mathbf R}^3$ with transition maps in the affine transformation group $Aff({\mathbf R}^3)$. Equivalently an affine $3$-manifold is a $3$-manifold with a flat torsion-free affine connection. We show that a closed affine $3$-mani…

The study of quotient structures in multi-graded bundles, including double vector bundles.

Paper classifies conic submanifolds in control systems.

Let X be a smooth, complete, connected submanifold of dimension n < N in a complex affine space A^N (C), and r is the rank of its Gauss map γ, γ(x) = T_x (X). The authors prove that if 2 \leq r \leq n - 1, N - n \geq 2, and in the pencil of the second fundamental forms of X, there are two forms defining a regular penci…

Extends PF submanifold results and connects Kac-Moody spaces.

New cubic forms linked to η-invariants and mod 2 indices.

The authors study the geometry of lightlike hypersurfaces on pseudo-Riemannian manifolds $(M, g)$ of Lorentzian signature. Such hypersurfaces are of interest in general relativity since they can be models of different types of physical horizons. For a lightlike hypersurface $V \subset (M, g)$ of general type and for so…

It is proved that the moduli space of all connected compact orientable embedded minimal affine Lagrangian submanifolds of a complex equiaffine space constitutes an infinite dimensional Frechet manifold (if it is not the empty set). The moduli space of all connected compact orientable metric Lagrangian embedded surfaces…

Extends Kähler metrics theory to symplectic manifolds with toric actions.

Simplified optimization for structured matrices in deep learning.

New framework for equivariant neural networks using Lie group decompositions.

A special linear Lie group over the real number field and the quarternion field admits a projectivley flat affine connection. We show that parabolic subgroups are autoparallel submanifolds and give a criterion the induced connection is projectively equivalent to a flat affine connection.

In this note we prove a Weierstrass representation formula for pluriminimal submanifolds of euclidean spaces. We use this formula to produce new families of examples of pluriminimal submanifolds. We also prove that any affine algebraic manifold can be pluriminimally embedded into some euclidean space in a non holomorph…

Proves algebraic cones for LCK manifolds with potential.

Parallel transport map over reductive spaces is an affine submersion.

Curves in higher dimensions are either affine or have super-Euclidean energy growth.

We introduce the new class of submanifolds of co-Banach type in tame Fréchet manifolds and construct tame Fréchet submanifolds as inverse images of regular values of certain tame maps. Our method furnishes an easy way to construct tame Fréchet manifolds. The results presented are key ingredients in the construction of …

Affine links in projective space have a specific group property.

Using standard methods for studying singularities of projections and of contacts, we classify the stable singularities of affine $λ$-equidistants of $n$-dimensional closed submanifolds of $\mathbb R^q$, for $q\leq 2n$, whenever $(2n,q)$ is a pair of nice dimensions.

The paper develops methods to generate invariant quantities in Metric-Affine Geometry.

Let Σbe a complete minimal Lagrangian submanifold of \C^n. We identify regions in the Grassmannian of Lagrangian subspaces so that whenever the image of the Gauss map of Σlies in one of these regions, then Σis an affine space.

The study translates Weinstein structures to Legendrian handlebodies for symplectic topology.

Invariants for virtual and twisted links using affine indices.

The abstract discusses transformations on statistical and semi-Weyl manifolds with torsion.