arXiv research

Geometric structures on surfaces relate to 2-plane distributions in 5D.

Geometric approach solves maximum likelihood for Cauchy-like distributions.

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

There is a remarkable type of field of two-planes special to four dimensions known as an Engel distributions. They are the only stable regular distributions besides the contact, quasi-contact and line fields. If an arbitrary two-plane field on a four-manifold is slightly perturbed then it will be Engel at generic point…

We study the sectional curvature of plane distributions on 3-manifolds. We show that if the distribution is a contact structure it is easy to manipulate this curvature. As a corollary we obtain that for every transversally oriented contact structure on a closed 3-dimensional manifold $M$ there is a metric, such that th…

By a real alphabeta-geometry we mean a four-dimensional manifold M equipped with a neutral metric h such that (M,h) admits both an integrable distribution of alpha-planes and an integrable distribution of beta-planes. We obtain a local characterization of the metric when at least one of the distributions is parallel (i…

The equitangent locus of a convex plane curve consists of the points from which the two tangent segments to the curve have equal length. The equitangent problem concerns the relation between the curve and its equitangent locus. An equitangent n-gon of a convex curve is a circumscribed n-gon whose vertices belong to the…

Generalizes Kauffman's clock theorem to surfaces.

In this paper, we propose a general model for plane-based clustering. The general model contains many existing plane-based clustering methods, e.g., k-plane clustering (kPC), proximal plane clustering (PPC), twin support vector clustering (TWSVC) and its extensions. Under this general model, one may obtain an appropria…

Random translation surfaces converge to a Poisson plane as genus grows.

In this note we announce some results, due to appear in [2], [3], on the structure of integral and normal currents, and their relation to Frobenius theorem. In particular we show that an integral current cannot be tangent to a distribution of planes which is nowhere involutive (Theorem 3.6), and that a normal current w…

Constructs polyhedral chains with prescribed tangent plane distributions.

Given a $C^1$ planes distribution $P_T$ on all ${\mathbb R}^m$ we consider {\em horizontal $α$-harmonic maps}, $α\ge 1/2$, with respect to such a distribution. These are maps $u\in H^α({\mathbb R}^k,{\mathbb R}^m)$ satisfying $P_T\nabla u=\nabla u$ and $P_T(u)(-Δ)^αu=0$ in ${\mathcal D}'({\mathbb R}^k).$ If the distrib…

Analyzes packing of circles in bounded and unbounded planes using mathematical formulas.

A beta function for double layers is defined and analyzed.

Study curvature of orthogonal distributions on manifolds.

The paper derives Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.

The study finds many tight contact structures on hyperbolic 3-spheres.

Two flat sub-Lorentzian problems on Martinet distribution differ in attainable set intersections.

As was shown recently by P. Nurowski, to any rank 2 maximally nonholonomic vector distribution on a 5-dimensional manifold M one can assign the canonical conformal structure of signature (3,2). His construction is based on the properties of the special 12-dimensional coframe bundle over M, which was distinguished by E.…

A new probabilistic polygonal curve representation using Gaussian Mixture Models.

We study the geometric properties of holomorphic distributions of totally null $m$-planes on a $(2m+ε)$-dimensional complex Riemannian manifold $(\mathcal{M}, \bm{g})$, where $ε\in {0,1}$ and $m \geq 2$. In particular, given such a distribution $\mathcal{N}$, say, we obtain algebraic conditions on the Weyl tensor and t…

We study the geometry and holonomy of semi-Riemannian, time-like metric cones that are indecomposable, i.e., which do not admit a local decomposition into a semi-Riemannian product. This includes irreducible cones, for which the holonomy can be classified, as well as non irreducible cones. The latter admit a parallel d…

Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either…

On a natural circle bundle T(M) over a 4-dimensional manifold M equipped with a split signature metric g, whose fibers are real totally null selfdual 2-planes, we consider a tautological rank 2 distribution D obtained by lifting each totally null plane horizontally to its point in the fiber. Over the open set where g i…

We consider the question whether an orientable 5-manifold can be equipped with a rank two distribution of Cartan type and what 2-plane bundles can be realized. We obtain a complete answer for open manifolds. In the closed case, we settle the topological part of this problem and present partial results concerning its ge…

Proposes a new complex Gaussian distribution for better modeling of complex-valued signals.

In spite of the recent surge of interest in quantile regression, joint estimation of linear quantile planes remains a great challenge in statistics and econometrics. We propose a novel parametrization that characterizes any collection of non-crossing quantile planes over arbitrarily shaped convex predictor domains in a…

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to $\mathbb{CP}^2$. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plan…

We define generalized distance-squared mappings, and we concentrate on the plane to plane case. We classify generalized distance-squared mappings of the plane into the plane in a recognizable way.

In this paper, the HyperKahler contact distribution of a 3-Sasakian manifold is studied. To analyze the curvature properties of this distribution, the special metric connection $\bar{\nabla}$ is defined. This metric connection is completely determined by HyperKahler contact distribution. We prove that HyperKahler conta…

The study classifies semiaffine stable planes into affine, projective, or punctured projective planes.

In the geometry of generic 2-plane fields on 5-manifolds, the local equivalence problem was solved by Cartan who also constructed the fundamental curvature invariant. For generic 2-plane fields or $(2,3,5)$-distributions determined by a single function of the form $F(q)$, the vanishing condition for the curvature invar…

Stable planes are locally isomorphic to classical projective planes.

Only vertical planes are asymptotic to other planes in 3D space.

It is known that for every smooth great circle fibration of the 3-sphere, the distribution of tangent 2-planes orthogonal to the fibres is a contact structure, in fact a tight one, but we show here that, beginning with the 5-sphere, there exist smooth great circle fibrations of all odd-dimensional spheres for which the…

Study of elementary planes in Apollonian orbifold with unusual equidistribution.

We show injectivity of the X-ray transform and the $d$-plane Radon transform for distributions on the $n$-torus, lowering the regularity assumption in the recent work by Abouelaz and Rouvière. We also show solenoidal injectivity of the X-ray transform on the $n$-torus for tensor fields of any order, allowing the tensor…

This paper continues a series of studies devoted to analysis of the bivariate probability distribution P(x,y) of two consecutive price increments x (push) and y (response) at intraday timescales for a group of stocks. Besides the asymmetry properties of P(x,y) such as Market Mill dependence patterns described in preced…

In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we relate this to the geometry of rank 2 distributions. We find a criterion for integration in quadratures and in closed fo…

Generic smooth plane-to-plane map germs are topologically equivalent to cones of mappings of the circle. We carry out a complete topological classification of smooth stable mappings of the circle and show how this classification leads, via the result mentioned above, to a topological classification of finitely determin…

Criteria for sharksfin and deltoid singularities from plane to plane, with applications.

We take the novel perspective to view data not as a probability distribution but rather as a current. Primarily studied in the field of geometric measure theory, $k$-currents are continuous linear functionals acting on compactly supported smooth differential forms and can be understood as a generalized notion of orient…

We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…

The study explores conformal planes with finite areas.

For a pair of points in a smooth locally convex surface in 3-space, its mid-plane is the plane containing its mid-point and the intersection line of the corresponding pair of tangent planes. In this paper we show that the limit of mid-planes when one point tends to the other along a direction is the Transon plane of th…

Analytic torsion defined for rank 2 distributions on 5-manifolds.

We construct a compact nonpositively curved squared 2-complex whose universal cover contains a flat plane that is not the limit of periodic flat planes.