Related papers: On Geodesic Completeness for Riemannian Metrics on…
This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the…
In this article, we study geometric aspects of semi-arithmetic Riemann surfaces by means of number theory and hyperbolic geometry. First, we show the existence of infinitely many semi-arithmetic Riemann surfaces of various shapes and prove…
Optimal transportation distances are valuable for comparing and analyzing probability distributions, but larger-scale computational techniques for the theoretically favorable quadratic case are limited to smooth domains or regularized…
Using quadratic forms, we stablish a criteria to relate the curvature of a Riemannian manifold and partial hyperbolicity of its geodesic flow. We show some examples which satisfy the criteria and another which does not satisfy it but still…
We start by constructing a Hilbert manifold T of orientation preserving diffeomorphisms of the circle (modulo the group of bi-holomorphic self-mappings of the disc). This space, which could be thought of as a completion of the universal…
Given a smooth compact surface without focal points and of higher genus, it is shown that its geodesic flow is semi-conjugate to a continuous expansive flow with a local product structure such that the semi-conjugation preserves…
The properties of geodesics flow are studied in a Friedmann-Robertson-Walker metric perturbed due to the inhomogeneities of matter. The basic, averaged Jacobi equation is derived, which reveals that the low density regions (voids) are able…
Shape spaces are fundamental in a variety of applications including image registration, morphing, matching, interpolation, and shape optimization. In this work, we consider two-dimensional shapes represented by triangular meshes of a given…
We introduce the geodesic walk for sampling Riemannian manifolds and apply it to the problem of generating uniform random points from polytopes in R^n specified by m inequalities. The walk is a discrete-time simulation of a stochastic…
This paper sheds light on the essential characteristics of geodesics, which frequently occur in considerations from motion in Euclidean space. Focus is mainly on a method of obtaining them from the calculus of variations, and an explicit…
We develop a geometric framework for irreversible transport phenomena in which macroscopic evolution equations arise from the combined structure of a thermodynamic state metric and an Onsager-based dissipation metric. The construction…
On the manifold $\Met(M)$ of all Riemannian metrics on a compact manifold $M$ one can consider the natural $L^2$-metric as described first by \cite{Ebin70}. In this paper we consider variants of this metric which in general are of higher…
We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed…
Using the works of Ma\~n\'e \cite{Ma} and Paternain \cite{Pat} we study the distribution of geodesic arcs with respect to equilibrium states of the geodesic flow on a closed manifold, equipped with a $\mathcal{C}^{\infty}$ Riemannian…
We provide an easy approach to the geodesic distance on the general linear group GL(n) for left-invariant Riemannian metrics which are also right-O(n)-invariant. The parametrization of geodesic curves and the global existence of length…
Insomuch as statistical mechanics circumvents the formidable task of addressing many-body dynamics, it remains a challenge to derive macroscopic properties from a solution to Hamiltonian equations for microscopic motion of an isolated…
We investigate the effect of different metrizations of probability spaces on the information geometric complexity of entropic motion on curved statistical manifolds. Specifically, we provide a comparative analysis based upon Riemannian…
In this paper we study the phenomenon of phase transitions for the geodesic flow on some geometrically finite negatively curved manifolds. We define a class of potentials going slowly to zero through the cusps of $M$ for which the pressure…
For random compositions of independent and identically distributed measurable maps on a Polish space, we study the existence and finitude of absolutely continuous ergodic stationary probability measures (which are, in particular, physical…
Uncertainty propagation and filtering can be interpreted as gradient flows with respect to suitable metrics in the infinite dimensional manifold of probability density functions. Such a viewpoint has been put forth in recent literature, and…