In this article, we study the geodesic problem in a generalized metric space, in which the distance function satisfies a relaxed triangle inequality $d(x,y)\leq σ(d(x,z)+d(z,y))$ for some constant $σ\geq 1$, rather than the usual triangle inequality. Such a space is called a quasimetric space. We show that many well-known results in metric spaces (e.g. Ascoli-Arzelà theorem) still hold in quasimetric spaces. Moreover, we explore conditions under which a quasimetric will induce an intrinsic metric. As an example, we introduce a family of quasimetrics on the space of atomic probability measures. The associated intrinsic metrics induced by these quasimetrics coincide with the $d_α$ metric studied early in the study of branching structures arisen in ramified optimal transportation. An optimal transport path between two atomic probability measures typically has a "tree shaped" branching structure. Here, we show that these optimal transport paths turn out to be geodesics in these intrinsic metric spaces.
We describe classes of coordinate systems in Carnot-Carathéodory spaces of low smoothness which allow for homogeneous approximations of quasimetrics and basis vector fields. We establish the minimal smoothness required for these classes to coinside with the class of the privileged coordinates described earlier for the smooth case. We also apply these results to prove partial analogues of existing results in the canonical coordinates of the 2nd kind. As a geometric tool we prove some convergence theorems in quasimetric spaces.
We obtained a complete classification of simple closed geodesics on regular tetrahedra in Lobachevsky space. Also, we evaluated the number of simple closed geodesics of length not greater than $L$ and found the asymptotic of this number as $L$ goes to infinity.
In this survey results on the behavior of simple closed geodesics on regular tetrahedra in three-dimensional spaces of constant curvature are presented.
We relate the existence of many infinite geodesics on Alexandrov spaces to a statement about the average growth of volumes of balls. We deduce that the geodesic flow exists and preserves the Liouville measure in several important cases. The developed analytic tool has close ties to integral geometry.
We define the notion of near geodesic between points of a metric space when no geodesic exists, and use this to extend Recio-Mitter's notion of geodesic complexity to non-geodesic spaces. This has potential application to topological robotics. We determine explicit near geodesics and geodesic complexity in a variety of cases.
In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales. We show that some such results remain valid for metric spaces with non-unique geodesic segments under suitable convexity assumptions on the distance function along distinguished geodesics. The discussion includes, among other things, the Flat Torus Theorem and Gromov's hyperbolicity criterion referring to embedded planes. This generalizes results of Bowditch for Busemann spaces.
We provide an alternative, constructive proof that the collection $\mathcal{M}$ of isometry classes of compact metric spaces endowed with the Gromov-Hausdorff distance is a geodesic space. The core of our proof is a construction of explicit geodesics on $\mathcal{M}$. We also provide several interesting examples of geodesics on $\mathcal{M}$, including a geodesic between $\mathbb{S}^0$ and $\mathbb{S}^n$ for any $n\geq 1$.
We have constructed a realization of rectilinear geodesic (in the sense of~\cite{Memoli2018}), lying in Gromov-Hausdorff space, as a shortest geodesic w.r.t. the Hausdorff distance in an ambient metric space.
We extend the classical result of Lyusternik and Fet on the existence of closed geodesics to singular spaces. We show that if $X$ is a compact geodesic metric space satisfying the CAT($κ$) condition for some fixed $κ>0$ and $π_n(X)\ne 0$ for some $n>0$ then $X$ has a periodic geodesic. This condition is satisfied for example by locally CAT($κ$) manifolds. Our result applies more generally to compact locally uniquely geodesic spaces.
In this paper we present a necessary conditions, that simple close geodesics on regular tetrahedra in the 3-dimensional hyperbolic space must satisfy. Furthermore, we explicitly describe three classes of simple closed geodesics on regular tetrahedra in the hyperbolic 3-space. These are so-called 2-homogeneous, 3-homogeneous and (3,2)-homogeneous geodesics. Up to a rigid motion of a tetrahedron there exists a unique geodesic in each class.
The collection $\mathcal{M}$ of all isometry classes of compact metric spaces endowed with the Gromov-Hausdorff distance $d_\mathcal{GH}$ is known to be a geodesic space. However, there is no known structural characterization of geodesics in $\mathcal{M}$. In this paper we provide two such characterizations. We first prove that every Gromov-Hausdorff geodesic is in fact a geodesic in the Hausdorff hyperspace of some compact metric space, which we call a Hausdorff geodesic. Inspired by this characterization, we further elucidate a structural connection between Hausdorff geodesics and Wasserstei…
We extend the concepts of de Casteljau and de Boor algorithms as well as splines to geodesic spaces and present some applications in geometric modeling. The concept of weighted geometric mean provides another approach to splines. We compare the corresponding Bezier curves and show that for Riemannian manifolds, their endpoint tangents coincide.
In this paper, we consider the Monge optimal transport problem with distance cost. We prove that in some metric spaces, possibly with many branching geodesics, an optimal transport map exists if the first marginal is absolutely continuous. The result is applicable to normed spaces and Hilbert geometries.
It is known that for a variety of choices of metrics, including the standard bottleneck distance, the space of persistence diagrams admits geodesics. Typically these existence results produce geodesics that have the form of a convex combination. More specifically, given two persistence diagrams and a choice of metric, one obtains a bijection realizing the distance between the diagrams, and uses this bijection to linearly interpolate from one diagram to another. We prove that for several families of metrics, every geodesic in persistence diagram space arises as such a convex combination. For ce…
For every proper geodesic space $X$ we introduce its quasi-geometric boundary $\partial_{QG}X$ with the following properties: 1. Every geodesic ray $g$ in $X$ converges to a point of the boundary $\partial_{QG}X$ and for every point $p$ in $\partial_{QG}X$ there is a geodesic ray in $X$ converging to $p$, 2. The boundary $\partial_{QG}X$ is compact metric, 3. The boundary $\partial_{QG}X$ is an invariant under quasi-isometric equivalences, 4. A quasi-isometric embedding induces a continuous map of quasi-geodesic boundaries, 5. If $X$ is Gromov hyperbolic, then $\partial_{QG}X$ is the Gromov bo…
