arXiv research

Computes a new metric quantity Y(M) for Riemannian 2d-manifolds.

Method finds domain of Laplace-Beltrami operator on 2D almost-Riemannian manifolds.

Study on a specific type of Riemannian manifolds constructed from 2D space-forms.

Researchers study Killing superalgebras in 2D manifolds.

Study characterizes kernel of mixed ray transform on simple surfaces.

New curvature K(x) measures manifold properties without integrals.

Study proves estimate for Hessian quotient equations on 2D Riemannian manifolds.

New method calculates cut locus on Riemannian manifolds using optimal transport.

The study finds conditions for a third rank Killing tensor field on a 2D Riemannian torus.

A 3D almost-Riemannian manifold is a generalized Riemannian manifold defined locally by 3 vector fields that play the role of an orthonormal frame, but could become collinear on some set $\Zz$ called the singular set. Under the Hormander condition, a 3D almost-Riemannian structure still has a metric space structure, wh…

A new privacy-preserving mechanism for shapes on manifolds.

Inspired by the Poisson Sigma Model and its relation to 2d gravity, we consider models governing morphisms from TSigma to any Lie algebroid E, where Sigma is regarded as d-dimensional spacetime manifold. We address the question of minimal conditions to be placed on a bilinear expression in the 1-form fields, S^ij(X) A_…

Researchers prove rigidity of 2D manifolds from boundary geodesic lengths.

In this article, we show how to embed the so-called CH2 equations into the geodesic flow of the Hdiv metric in 2D, which, itself, can be embedded in the incompressible Euler equation of a non compact Riemannian manifold. The method consists in embedding the incompressible Euler equation with a potential term coming fro…

Study of symmetries in a 2D space with specific metric properties.

New theorem shows embedding restrictions for manifold skeletons.

In this paper we prove a sub-Riemannian version of the classical Santaló formula: a result in integral geometry that describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. Our construction works under quite general assumptions, satisfied by any sub-Riemannian structure ass…

A 2D Riemannian space has only 2 injective geodesics.

Study on rolling of 2D and 3D manifolds, identifying orbit dimensions.

In 2D, a unique solution is proven for a specific equation.

2d dualities linked to 4-simplex triangulation.

The paper approximates Levi-Civita connection and curvature on 2D manifolds using finite elements.

We describe rules for building 2d theories labeled by 4-manifolds. Using the proposed dictionary between building blocks of 4-manifolds and 2d N=(0,2) theories, we obtain a number of results, which include new 3d N=2 theories T[M_3] associated with rational homology spheres and new results for Vafa-Witten partition fun…

Derive bihamiltonian structure for rational reduction of 2D-Toda hierarchy

Study bihamiltonian structures and Frobenius manifolds for specific Toda hierarchies.

This article deals with 2d almost Riemannian structures, which are generalized Riemannian structures on manifolds of dimension 2. Such sub-Riemannian structures can be locally defined by a pair of vector fields (X,Y), playing the role of orthonormal frame, that may become colinear on some subset. We denote D = span(X,Y…

We prove a "gluing" theorem for monotone homotopies; a monotone homotopy is a homotopy through simple contractible closed curves which themselves are pairwise disjoint. We show that two monotone homotopies which have appropriate overlap can be replaced by a single monotone homotopy. The ideas used to prove this theorem…

Study third order differential operators on 2D manifolds, finding equivalence conditions.

Geometric interpretation of 2d-4d wall-crossing formulas.

A simple Almost-Riemmanian Structure on a Lie group G is defined by a linear vector field and dim(G)-1 left-invariant ones. We state results about the singular locus, the abnormal extremals and the desingularization of such ARS's, and these results are illustrated by examples on the 2D affine and the Heisenberg groups.…

The study characterizes constant curvature manifolds using ruled surfaces.

This paper deals with the computation of sectional curvature for the manifolds of $N$ landmarks (or feature points) in D dimensions, endowed with the Riemannian metric induced by the group action of diffeomorphisms. The inverse of the metric tensor for these manifolds (i.e. the cometric), when written in coordinates, i…

The paper establishes bounds on the lengths of geodesics on manifolds with curvature constraints.

The paper finds a family of 2D surfaces where Laplacian and Dirichlet-to-Neumann commute.

The Grove-Searle theorem on 2d manifolds with 8 or less symmetry groups has positive Euler characteristic.

Connects 4-manifold topology to topological modular forms.

In this paper we study geometries on the manifold of curves. We define a manifold $M$ where objects $c\in M$ are curves, which we parameterize as $c:S^1\to \real^n$ ($n\ge 2$, $S^1$ is the circle). Given a curve $c$, we define the tangent space $T_cM$ of $M$ at $c$ including in it all deformations $h:S^1\to\real^n$ of …

We answer a question of Liokumovich-Nabutovsky-Rotman showing that if D is a Riemannian 2-disc with boundary length L, diameter d and area A << d then D can be filled by a homotopy where the lengths of the intermediate curves are bounded by $L+2d+O(\sqrt A)$.

The study constructs AdS manifolds from Gromov-Thurston manifolds.

A novel method compresses point cloud attributes by folding them onto a 2D grid.

Study geodesic ray transform on 2D manifolds with conjugate points.

For integers $d \geq 2$ and $ε= 0$ or 1, let $S^{1, d - 1}(ε)$ denote the sphere product $S^{1} \times S^{d - 1}$ if $ε= 0$ and the twisted $S^{d - 1}$ bundle over $S^{1}$ if $ε= 1$. The main results of this paper are: (a) if $d \equiv ε$ (mod 2) then $S^{1, d - 1}(ε)$ has a unique minimal triangulation using $2d + 3$ …

Following the previous authors works (joint with I.A.Dynnikov) we develop a theory of the discrete analogs of the differential-geometrical (DG) connections in the triangulated manifolds. We study a nonstandard discretization based on the interpretation of DG Connection as linear first order (''triangle'') difference eq…

We consider 3D flow equations inspired by the renormalization group (RG) equations of string theory with a three dimensional target space. By modifying the flow equations to include a U(1) gauge field, and adding carefully chosen De Turck terms, we are able to extend recent 2D results of Bakas to the case of a 3D Riema…

Study of vortex interactions in Ginzburg-Landau models on 2D Riemannian manifolds.

Proposes TS-NMF for 2D clustering, preserving spatial info.

We generalize and strengthen the theorem of Gromov that every compact Riemannian manifold of diameter at most D has a set of generators g_1,...,g_k of length at most 2D and relators of the form g_ig_m = g_j . In particular, we obtain an explicit bound for the number k of generators in terms of the number "short loops" …

The paper develops a theory linking Hamiltonian and quasi-Hamiltonian manifolds.