sub-Riemannian Geometry Research Articles

Mahler type theorem and non-collapsed limits of sub-Riemannian compact Heisenberg manifolds

Mahler type theorem and non-collapsed limits of sub-Riemannian compact Heisenberg manifolds

Computing Euclidean distance and maximum likelihood retraction maps for constrained optimization

Riemannian optimization uses local methods to solve optimization problems whose constraint set is a smooth manifold. A linear step along some descent direction usually leaves the constraints, and hence retraction maps are used to approximate the exponential map and return to the manifold. For many common matrix manifolds, retraction maps are available, with more or less explicit formulas. For implicitly-defined manifolds, suitable retraction maps are difficult to compute. We therefore develop an algorithm which uses homotopy continuation to compute the Euclidean distance retraction for any implicitly-defined submanifold of Rn, and prove convergence results.We also consider statistical models as Riemannian submanifolds of the probability simplex with the Fisher metric. Replacing Euclidean distance with maximum likelihood results in a map which we prove is a retraction. In fact, we prove the retraction is second-order; with the Levi-Civita connection associated to the Fisher metric, it approximates geodesics to second-order accuracy.

Large Deviations for Small Noise Hypoelliptic Diffusion Bridges on Sub-Riemannian Manifolds

In this paper we study a large deviation principle of Freidlin–Wentzell type for pinned hypoelliptic diffusion measures associated with a natural sub-Laplacian on a compact sub-Riemannian manifold. To prove this large deviation principle, we use rough path theory and manifold-valued Malliavin calculus.

Kernel density estimation for a stochastic process with values in a Riemannian manifold

This paper is related to the issue of the density estimation of observations with values in a Riemannian submanifold. In this context, Henry and Rodriguez ((2009), ‘Kernel Density Estimation on Riemannian Manifolds: Asymptotic Results’, Journal of Mathematical Imaging and Vision, 34, 235–239) proposed a kernel density estimator for independent data. We investigate here the behaviour of Pelletier's estimator when the observations are generated from a strictly stationary α-mixing process with values in this submanifold. Our study encompasses both pointwise and uniform analyses of the weak and strong consistency of the estimator. Specifically, we give the rate of convergence in terms of mean square error, probability, and almost sure convergence (a.s.). We also give a central-limit theorem and illustrate our proposal through some simulations and a real data application.

A DDVV Conjecture for Riemannian Maps

The Wintgen inequality is a significant result in the field of differential geometry, specifically related to the study of submanifolds in Riemannian manifolds. It was discovered by Pierre Wintgen. In the present work, we deal with the Riemannian maps between Riemannian manifolds that serve as a superb method for comparing the geometric structures of the source and target manifolds. This article is the first to explore a well-known conjecture, called DDVV inequality (a conjecture for Wintgen inequality on Riemannian submanifolds in real space forms proven by P.J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken), for Riemannian maps, where we consider different space forms as target manifolds. There are numerous research problems related to such inequality in various ambient manifolds. These problems can all be explored within the general framework of Riemannian maps between various Riemannian manifolds equipped with notable geometric structures.

The Relative Heat Content for Submanifolds in Sub-Riemannian Geometry

The Relative Heat Content for Submanifolds in Sub-Riemannian Geometry

On the asymptotic behaviour of the fractional Sobolev seminorms: A geometric approach

We study the well-known asymptotic formulas for fractional Sobolev functions à la Bourgain–Brezis–Mironescu and Maz'ya–Shaposhnikova, in a geometric approach. We show that the key to these asymptotic formulas are Rademacher's theorem and volume growth at infinity respectively. Examples fitting our framework includes Euclidean spaces, Riemannian manifolds, Alexandrov spaces, finite dimensional Banach spaces, and some ideal sub-Riemannian manifolds.

Overdetermined problems in groups of Heisenberg type: Conjectures and partial results

In this paper we formulate some conjectures in sub-Riemannian geometry concerning a characterisation of the Koranyi-Kaplan ball in a group of Heisenberg type through the existence of a solution to suitably overdetermined problems. We prove an integral identity that provides a rigidity constraint for one of the two problems. By exploiting some new invariances of these Lie groups, for domains having partial symmetry we solve these problems by converting them to known results for the classical p-Laplacian.

Collapsed limits of compact Heisenberg manifolds with sub-Riemannian metrics

Collapsed limits of compact Heisenberg manifolds with sub-Riemannian metrics

Sub-Riemannian Geometry of Curves and Surfaces in Roto-Translation Group Associated with Canonical Connection

The aim of this paper is to obtain the sub-Riemannian properties of the roto-translation group RT. At the same time, we compute the sub-Riemannian limits of Gaussian curvature associated with two kinds of canonical connections for a C2-smooth surface in the roto-translation group away from characteristic points and signed geodesic curvature associated with two kinds of canonical connections for C2-smooth curves on surfaces. Based on these results, we obtain a Gauss-Bonnet theorem in the RT.

Biconservative Riemannian Submanifolds in Minkowski 5-Space

In this study, we focus biconservative Riemannian submanifolds with parallel normalized mean curvature vector field (PNMCV) in $\mathbb{E}^5_1$. We obtain explicit classifications for these submanifolds with exactly two distinct principal curvatures of the shape operator along the mean curvature vector field. In particular, we investigate these submanifolds which have time-like and space-like mean curvature vector field in $\mathbb{E}^5_1$.

Space-only gradient estimates of Schrödinger equation with Neumann boundary condition under integral Ricci curvature bounds

Assume that M (M⊆N) is an n-dimensional compact Riemannian submanifold with boundary, satisfying the integral Ricci curvature assumption:D2supx∈N⁡(∮B(x,d)|Ric−|pdy)1p<κ for κ=κ(n,p) small enough, p>n/2, where diam(M)≤D. The boundary of M needs to satisfy the interior rolling R-ball condition. We prove a Hamilton type gradient estimate for positive solutions to the parabolic Schrödinger equation with Neumann boundary conditions, on a compact Riemannian submanifold M with boundary ∂M, satisfying the integral Ricci curvature assumption.

Symplectic reduction of the sub-Riemannian geodesic flow for metabelian nilpotent groups

We consider nilpotent Lie groups for which the derived subgroup is abelian. We equip them with sub-Riemannian metrics and we study the normal Hamiltonian flow on the cotangent bundle. We show a correspondence between normal trajectories and polynomial Hamiltonians in some Euclidean space. We use the aforementioned correspondence to give a criterion for the integrability of the normal Hamiltonian flow. As an immediate consequence, we show that in Engel-type groups the flow of the normal Hamiltonian is integrable. For Carnot groups that are semidirect products of two abelian groups, we give a set of conditions that normal trajectories must fulfill to be globally length-minimizing. Our results are based on a symplectic reduction procedure.

Homogeneous Sub-Riemannian Manifolds Whose Normal Extremals are Orbits

Homogeneous Sub-Riemannian Manifolds Whose Normal Extremals are Orbits

Geometry of sub - Riemannian manifolds equipped with a semimetric quarter - symmetric connection

On a sub-Riemannian manifold we introduce a semimetric quarter-symmetric connection by defining intrinsic metric connection and two structural endomorphisms preserving the distribution on a sub-Riemannian manifold. We find conditions ensuring the metric property of the introduced connection. We clarify the nature of the structural endomorphisms of semimetric connection consistent with a sub-Riemannian quasi-static structure defined on non-holonomic Kenmotsu manifold and on almost quasi-Sasakian manifold. We find conditions, under which the mentioned manifolds are Einstein manifolds with respect to the quarter-symmetric connection.

Stability of minimal surfaces in the sub-Riemannian manifold $\widetilde{E(2)}$

In the paper we study smooth oriented surfaces in the universal covering space of the group of orientation-preserving Euclidean plane isometries, which has a three-dimensional sub-Riemannian manifold structure. This structure is constructed as a restriction of the Euclidean metric on the group to some completely non-integrable left invariant distribution. The sub-Riemannian area of a surface is then defined as the integral of the length of its unit normal field projected orthogonally onto this distribution. We calculate the first variation formula of the sub-Riemannian surface area and derive the minimality criterion from it. Here we call a surface minimal if it is a critical point of the sub-Riemannian area functional under normal variations with compact support. We show that the minimality in this case is not equivalent to the vanishing of the sub-Riemannian mean curvature. We then prove that a Euclidean plane is minimal if and only if it is parallel or orthogonal to the $z$-axis (where the $z$-coordinate corresponds to the rotation angle of an isometry). Also we obtain the minimality condition for a graph and give examples of minimal graphs. The examples considered in the paper demonstrate, in particular, that the minimality of a surface in the Riemannian (in this case Euclidean) sense does not imply its sub-Riemannian minimality, and vice versa. Next, we consider the stability of minimal surfaces. For this purpose, we derive the second variation formula of the sub-Riemannian area and show with it that minimal Euclidean planes are stable. We introduce a class of surfaces for which the tangent planes are perpendicular to the planes of the sub-Riemannian structure, and call them vertical surfaces. In particular, for such surfaces the second variation formula is simplified significantly. Then we prove that complete connected vertical minimal surfaces are either Euclidean planes or helicoids and that helicoids are unstable. This implies a following Bernstein type result: a complete connected vertical minimal surface is stable if and only if it is a Euclidean plane orthogonal to the $z$-axis.

Geodesic Transformations of Distributions of Sub-Riemannian Manifolds

Geodesic Transformations of Distributions of Sub-Riemannian Manifolds

Symmetrized and non-symmetrizedasymptotic mean value Laplacian in metric measure spaces

The asymptotic mean value Laplacian—AMV Laplacian—extends the Laplace operator from $\mathbb {R}^n$ to metric measure spaces through limits of averaging integrals. The AMV Laplacian is however not a symmetric operator in general. Therefore, we consider a symmetric version of the AMV Laplacian, and focus lies on when the symmetric and non-symmetric AMV Laplacians coincide. Besides Riemannian and 3D contact sub-Riemannian manifolds, we show that they are identical on a large class of metric measure spaces, including locally Ahlfors regular spaces with suitably vanishing distortion. In addition, we study the context of weighted domains of $\mathbb {R}^n$ —where the two operators typically differ—and provide explicit formulae for these operators, including points where the weight vanishes.

A Universal Heat Semigroup Characterisation of Sobolev and BV Spaces in Carnot Groups

Abstract In sub-Riemannian geometry there exist, in general, no known explicit representations of the heat kernels, and these functions fail to have any symmetry whatsoever. In particular, they are not a function of the control distance, nor they are for instance spherically symmetric in any of the layers of the Lie algebra. Despite these unfavourable aspects, in this paper we establish a new heat semigroup characterisation of the Sobolev and $BV$ spaces in a Carnot group by means of an integral decoupling property of the heat kernel.

Local non-injectivity of the exponential map at critical points in sub-Riemannian geometry

We prove that the sub-Riemannian exponential map is not injective in any neighbourhood of certain critical points. Namely that it does not behave like the injective map of reals given by f(x)=x3 near its critical point x=0. As a consequence, we characterise conjugate points in ideal sub-Riemannian manifolds in terms of the metric structure of the space. The proof uses the Hilbert invariant integral of the associated variational problem.