Geodesic Space Research Articles

Resolvents of equilibrium problems on a complete geodesic space with curvature bounded above

We consider equilibrium problems on a complete geodesic space with curvature bounded above by one and propose the notion of resolvents for this problem. We prove its well-definedness as a single-valued mapping whose domain is the whole space, and its geometric properties.

Supervised learning with indefinite topological Kernels

Topological Data Analysis (TDA) is a new branch of statistics devoted to the study of the ‘shape’ of the data. As TDA's tools are typically defined in complex spaces, kernel methods are often used to perform inferential task by implicitly mapping topological summaries, most noticeably the Persistence Diagram (PD), to vector spaces. For positive definite kernels defined on PDs, however, kernel embeddings do not fully retain the metric structure of the original space. We introduce a new exponential kernel, built on the geodesic space of PDs, and we show with simulated and real applications how it can be successfully used in regression and classification tasks, despite not being positive definite.

Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition

Abstract We study the interaction between the g.o. property and certain flag curvature conditions. A Finsler manifold is called g.o. if each constant speed geodesic is the orbit of a one-parameter subgroup. Besides the non-negatively curved condition, we also consider the condition (FP) for the flag curvature, i.e. in any flag we find a flag pole such that the flag curvature is positive. By our main theorem, if a g.o. Finsler space (M, F) has non-negative flag curvature and satisfies (FP), then M is compact. If M = G/H where G has a compact Lie algebra, then the rank inequality rk 𝔤 ≤ rk 𝔥+1 holds. As an application,we prove that any even-dimensional g.o. Finsler space which has non-negative flag curvature and satisfies (FP) is a smooth coset space admitting a positively curved homogeneous Riemannian or Finsler metric.

A novel iterative approach for solving common fixed point problems in Geodesic spaces with convergence analysis

In this paper, we introduce a new iterative method for nonexpansive mappings in CAT(\kappa) spaces. First, the rate of convergence of proposed method and comparison with recently existing method is proved. Second, strong and \Delta-convergence theorems of the proposed method in such spaces under some mild conditions are also proved. Finally, we provide some non-trivial examples to show efficiency and comparison with many previously existing methods.

Some Continuation results in Uniquely Geodesic Spaces

In this paper, we prove some continuation results based on essential maps in −uniquely geodesic spaces, which are generalizations of the existing results in Banach spaces. Also, we obtain Leray Schauder principle in −uniquely geodesic spaces which generalizes Leray Schauder principle in the setting of Hyperbolic spaces.

On PSL(2,ℂ) and on the space of geodesics of ℍ 3 as holomorphic Riemannian manifolds

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Wolff–Denjoy theorems in geodesic spaces

We show a Wolff-Denjoy type theorem in complete geodesic spaces in the spirit of Beardon's framework that unifies several results in this area. In particular, it applies to strictly convex bounded domains in $\mathbb{R}^{n}$ or $\mathbb{C}^{n}$ with respect to a large class of metrics including Hilbert's and Kobayashi's metrics. The results are generalized to $1$-Lipschitz compact mappings in infinite-dimesional Banach spaces.

A weak notion of visibility, a family of examples, and Wolff-Denjoy theorems

We investigate a form of visibility introduced recently by Bharali and Zimmer -- and shown to be possessed by a class of domains called Goldilocks domains. The range of theorems established for these domains stem from this form of visibility together with certain quantitative estimates that define Goldilocks domains. We show that some of the theorems alluded to follow merely from the latter notion of visibility. We call those domains that possess this property visibility domains with respect to the Kobayashi distance. We provide a sufficient condition for a domain in $\mathbb{C}^n$ to be a visibility domain. A part of this paper is devoted to constructing a family of domains that are visibility domains with respect to the Kobayashi distance but are not Goldilocks domains. Our notion of visibility is reminiscent of uniform visibility in the context of CAT(0) spaces. However, this is an imperfect analogy because, given a bounded domain $\Omega$ in $\mathbb{C}^n$, $n\geq 2$, it is, in general, not even known whether the metric space $(\Omega,{\sf k}_{\Omega})$ (where ${\sf k}_{\Omega}$ is the Kobayashi distance) is a geodesic space. Yet, with just this weak property, we establish two Wolff--Denjoy-type theorems.

Gromov’s Amenable Localization and Geodesic Flows

Let M be a compact smooth Riemannian n-manifold with boundary. We combine Gromov’s amenable localization technique with the Poincaré duality to study the traversally generic geodesic flows on SM, the space of the spherical tangent bundle. Such flows generate stratifications of SM, governed by rich universal combinatorics. The stratification reflects the ways in which the flow trajectories are tangent to the boundary \(\partial (SM)\). Specifically, we get lower estimates of the numbers of connected components of these flow-generated strata of any given codimension k in terms of the normed homology \(H_k(M; \mathbb R)\) and \(H_k(DM; \mathbb R)\), where \(DM = M\cup _{\partial M} M\) denotes the double of M. The norms here are the simplicial semi-norms in homology. The more complex the metric on M is, the more numerous the strata of SM and S(DM) are. It turns out that the normed homology spaces form obstructions to the existence of globally k-convex traversally generic metrics on M. We also prove that knowing the geodesic scattering map on M makes it possible to reconstruct the stratified topological type of the space of geodesics, as well as the amenably localized Poincaré duality operators on SM.

Some results on Best proximity in geodesic spaces

Eldred and Veeramani have proved the existence and uniqueness of best proximity for contraction cyclic contraction mapping in uniformly convex banach spaces. In this paper, Eldred and Veeramani study is extended to geodesic spaces particularly in CAT(0) spaces, prove the sequences (x_2n) and (x_2n+1) are Cauchy sequences and prove the existenace and uniqueness of best proximity point for contraction cyclic mapping in CAT(0) spaces.

Iterative Sequences for a Finite Number of Resolvent Operators on Complete Geodesic Spaces

We consider Halpern’s and Mann’s types of iterative schemes to find a common minimizer of a finite number of proper lower semicontinuous convex functions defined on a complete geodesic space with curvature bounded above.

A uniform betweenness property in metric spaces and its role in the quantitative analysis of the “Lion-Man” game

In this paper we analyze, based on an interplay between ideas and techniques from logic and geometric analysis, a pursuit-evasion game. More precisely, we focus on a uniform betweenness property and use it in the study of a discrete lion and man game with an $\varepsilon$-capture criterion. In particular, we prove that in uniformly convex bounded domains the lion always wins and, using ideas stemming from proof mining, we extract a uniform rate of convergence for the successive distances between the lion and the man. As a byproduct of our analysis, we study the relation among different convexity properties in the setting of geodesic spaces.

The contact structure on the space of null geodesics of causally simple spacetimes

It is shown that the space of null geodesics of a star-shaped causally simple subset of Minkowski space is contactomorphic to the canonical contact structure in the spherical cotangent bundle of Rn. In the 3-dimensional case we prove a similar result for a large class of causally simple contractible subsets of an arbitrary globally hyperbolic spacetime applying methods from the theory of contact-convex surfaces. Moreover we prove that under certain assumptions the space of null geodesics of a causally simple spacetime embeds with smooth boundary into the space of null geodesics of a globally hyperbolic spacetime. The characteristic foliation of this boundary provides an invariant of the conformal class of the causally simple spacetime.

Measure Theoretic Entropy of Discrete Geodesic Flow on Nagao Lattice Quotient.

The discrete geodesic flow on Nagao lattice quotient of the space of bi-infinite geodesics in regular trees can be viewed as the right diagonal action on the double quotient of by and . We investigate the measure-theoretic entropy of the discrete geodesic flow with respect to invariant probability measures.

Analytical Solution for Sparse Data Interpolation Using Geodesic Distance Affinity Space: Application to the Optical Flow Problem and 3D Reconstruction

In this paper, we derive the analytic solution for sparse data interpolation using the geodesic distance affinity space of the known reference image associated with sparse data to be interpolated. We compare our method with the EpicFlow algorithm [1] that is intuitively motivated by almost the same geodesic distance principle. However, we found that our approach is more general, faster, and with clearer theoretical motivation. To test the accuracy of our approach, we applied our interpolation method to the sparse optical flow data obtained by the DCflow convolution neural network method [2] and compared our result with the EpicFlow interpolation result on the same sparse data set. The comparison shows that our algorithm is more accurate than the EpicFlow technique.

On Solving Minimization Problem and Common Fixed Point Problem Over Geodesic Spaces With Curvature Bounded Above

In this paper, we introduce a new modified proximal point algorithm for solving minimization problems and common fixed point problem in CAT\((1)\) spaces. We prove strong and \(\Delta\)-convergence theorems under some mild conditions. Further, an application on convex minimization and common fixed point problem over CAT\((\kappa)\) spaces with the bounded positive real number \(\kappa\) are presented. Our results extend and improve the corresponding recent results in the literature.

Unit Ball Graphs on Geodesic Spaces

Consider finitely many points in a geodesic space. If the distance of two points is less than a fixed threshold, then we regard these two points as "near". Connecting near points with edges, we obtain a simple graph on the points, which is called a unit ball graph. If the space is the real line, then it is known as a unit interval graph. Unit ball graphs on a geodesic space describe geometric characteristics of the space in terms of graphs. In this article, we show that every unit ball graph on a geodesic space is (strongly) chordal if and only if the space is an $ \mathbb{R} $-tree and that every unit ball graph on a geodesic space is (claw, net)-free if and only if the space is a connected manifold of dimension at most $ 1 $. As a corollary, we prove that the collection of unit ball graphs essentially characterizes the real line and the unit circle.

Equivalence of ray monotonicity properties and classification of optimal transport maps for strictly convex norms

Abstract In this paper, we first define ray increasing and decreasing monotonicity of maps. If 𝑇 is an optimal transport map for the Monge problem with cost function ∥ y - x ∥ sc \lVert y-x\rVert_{\mathrm{sc}} in R n R^{n} or 𝑇 is an optimal transport map for the Monge problem with cost function d ⁢ ( x , y ) d(x,y) , the geodesic distance, in more general, non-branching geodesic spaces 𝑋, we show respectively equivalence of some previously introduced monotonicity properties and the property of ray increasing as well as ray decreasing monotonicity which we define in this paper. Then, by solving secondary variational problems associated with strictly convex and concave functions respectively, we show that there exist ray increasing and decreasing optimal transport maps for the Monge problem with cost function ∥ y - x ∥ sc \lVert y-x\rVert_{\mathrm{sc}} . Finally, we give the classification of optimal transport maps for the Monge problem such that the cost function ∥ y - x ∥ sc \lVert y-x\rVert_{\mathrm{sc}} further satisfies the uniform smoothness and convexity estimates. That is, all of the optimal transport maps for such Monge problem can be divided into three different classes: the ray increasing map, the ray decreasing map and others.

Gödel spacetime, planar geodesics and the Möbius map

Timelike geodesics on a hyperplane orthogonal to the symmetry axis of the Gödel spacetime appear to be elliptic-like if standard coordinates naturally adapted to the cylindrical symmetry are used. The orbit can then be suitably described through an eccentricity-semi-latus rectum parametrization, familiar from the Newtonian dynamics of a two-body system. However, changing coordinates such planar geodesics all become explicitly circular, as exhibited by Kundt’s form of the Gödel metric. We derive here a one-to-one correspondence between the constants of the motion along these geodesics as well as between the parameter spaces of elliptic-like versus circular geodesics. We also show how to connect the two equivalent descriptions of particle motion by introducing a pair of complex coordinates in the 2-planes orthogonal to the symmetry axis, which brings the metric into a form which is invariant under Möbius transformations preserving the symmetries of the orbit, i.e., taking circles to circles.

Some Results On Fixed Points For Weakly Inward Mappings In Geodesic Metrics Paces

Geodesic spaces are convex nonlinear spaces. Convexity is a significant tool to generalize some properties of Banach spaces. In this paper, the characterization of weakly inward was extended to CAT(0) spaces and give equivalent condition for the existence of fixed point for multivalued mapping