Related papers: Geodesics on Grushin spaces

We introduce Gaussian quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices. By definition, these spaces are of even degrees. The optimal quadrature rules we recently…

In this paper we extend the concept of a conjugate point in a Riemannian manifold to complete length spaces (also known as geodesic spaces). In particular, we introduce symmetric conjugate points and ultimate conjugate points. We then…

We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most $L$ on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the…

We propose a globally-accelerated, first-order method for the optimization of smooth and (strongly or not) geodesically-convex functions in a wide class of Hadamard manifolds. We achieve the same convergence rates as Nesterov's accelerated…

We take a three dimensional Euclidian metric in toroidal coordinates and consider the corresponding Laplace equation. The simplest solution of this equation is taken. Based on this we build a Weyl space-time. This space-time is transformed…

In this paper we present two frameworks in which global maximization of a bounded hessian function over a strongly convex set can be reduced to convex optimization. The first presented framework is a continuation of one of our previous…

We analyze a variational time discretization of geodesic calculus on finite- and certain classes of infinite-dimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete…

This letter describes a novel derivation of general relativity by considering the (non)self-consistency of theories whose Hamiltonians are constraints. The constraints, from Hamilton's equations, generate the evolution, while the evolution,…

The starting point of this work is the principle that all movement of particles and photons must follow geodesics of a 4-dimensional space where time intervals are always a measure on geodesic arc lengths. The last 3 coordinates (alpha =…

We consider geodesically convex optimization problems involving distances to a finite set of points $A$ in a CAT(0) cubical complex. Examples include the minimum enclosing ball problem, the weighted mean and median problems, and the…

We consider the problem of recovering an unknown matching between a set of $n$ randomly placed points in $\mathbb{R}^d$ and random perturbations of these points. This can be seen as a model for particle tracking and more generally, entity…

The theory of geodesic regression aims to find a geodesic curve which is an optimal fit to a given set of data. In this article we restrict ourselves to the Riemannian manifold of positive definite operators (matrices) on a Hilbert space of…

We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on $SL(2, \mathbb{R})$. Our method uses a symmetry reduction which translates the problem into a Riemannian problem on a two…

Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers' theorem gives a global bound on the length of the first $3g-3$ geodesics. We use the construction of Brooks and Makover of random Riemann…

We consider the space of probability measures on a discrete set $X$, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset $Y \subseteq X$, it is natural to ask whether they can be connected…

The action of the mapping class group of the thrice-punctured projective plane on its $\mathrm{GL}(2,\mathbb{C})$ character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective…

Let $\mathcal{H}$ be a reproducing kernel Hilbert space of functions on a set $X$. We study the problem of finding a minimal geodesic of the Grassmann manifold of $\mathcal{H}$ that joins two subspaces consisting of functions which vanish…

We give a universal upper bound for the total curvature of minimizing geodesic on a convex surface in the Euclidean space.

We develop a convergence theory of space-time discretizations for the linear, 2nd-order wave equation in polygonal domains $\Omega\subset\mathbb{R}^2$, possibly occupied by piecewise homogeneous media with different propagation speeds.…

We prove sectional and Ricci-type comparison theorems for the existence of conjugate points along sub-Riemannian geodesics. In order to do that, we regard sub-Riemannian structures as a special kind of variational problems. In this setting,…