Geodesic Space Research Articles

Isometric rigidity of Wasserstein spaces over Euclidean spheres

We study the structure of isometries of the quadratic Wasserstein space W2(Sn,‖⋅‖) over the sphere endowed with the distance inherited from the norm of Rn+1. We prove that W2(Sn,‖⋅‖) is isometrically rigid, meaning that its isometry group is isomorphic to that of (Sn,‖⋅‖). This is in striking contrast to the non-rigidity of its ambient space W2(Rn+1,‖⋅‖) but in line with the rigidity of the geodesic space W2(Sn,∢). One of the key steps of the proof is the use of mean squared error functions to mimic displacement interpolation in W2(Sn,‖⋅‖). A major difficulty in proving rigidity for quadratic Wasserstein spaces is that one cannot use the Wasserstein potential technique. To illustrate its general power, we use it to prove isometric rigidity of Wp(S1,‖⋅‖) for 1≤p<2.

Convergence theorems of implicit type iterations in geodesic spaces with negative curvature

Convergence theorems of implicit type iterations in geodesic spaces with negative curvature

Persistent Homology with Selective Rips Complexes Detects Geodesic Circles

This paper introduces a method to detect each geometrically significant loop that is a geodesic circle (an isometric embedding of S1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S^1$$\\end{document}) and a bottleneck loop (meaning that each of its perturbations increases the length) in a geodesic space using persistent homology. Under fairly mild conditions, we show that such a loop either terminates a 1-dimensional homology class or gives rise to a 2-dimensional homology class in persistent homology. The main tool in this detection technique are selective Rips complexes, new custom made complexes that function as an appropriate combinatorial lens for persistent homology to detect the above mentioned loops. The main argument is based on a new concept of a local winding number, which turns out to be an invariant of certain homology classes.

Hölder Parameterization of Continuous Quasi-Self-Contracted Curves in Complete Geodesic Spaces

Hölder Parameterization of Continuous Quasi-Self-Contracted Curves in Complete Geodesic Spaces

On some results related to the Karlsson–Nussbaum conjecture in geodesic spaces

We show a Wolff–Denjoy type theorem in the case of a one-parameter continuous semigroup of nonexpansive mappings in which there is a compact mapping. Using the notion of attractor, we are also able to prove some specific properties directly related to the Karlsson–Nussbaum conjecture.

Fundamental groups and group presentations with bounded relator lengths

We study the geometry of compact geodesic spaces with trivial first Betti number admitting large finite groups of isometries. We show that if a finite group G acts by isometries on a compact geodesic space X whose first Betti number vanishes, then diam(X)/diam(X/G)≤4|G|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {diam}}(X) / {\ ext {diam}}(X / G ) \\le 4 \\sqrt{ \\vert G \\vert }$$\\end{document}. For a group G and a finite symmetric generating set S, Pk(Γ(G,S))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_k(\\varGamma (G, S))$$\\end{document} denotes the 2-dimensional CW-complex whose 1-skeleton is the Cayley graph Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varGamma $$\\end{document} of G with respect to S and whose 2-cells are m-gons for 0≤m≤k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 \\le m \\le k$$\\end{document}, defined by the simple graph loops of length m in Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varGamma $$\\end{document}, up to cyclic permutations. Let G be a finite abelian group with |G|≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\vert G \\vert \\ge 3$$\\end{document} and S a symmetric set of generators for which Pk(Γ(G,S))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_k(\\varGamma (G,S))$$\\end{document} has trivial first Betti number. We show that the first nontrivial eigenvalue -λ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\\lambda _1$$\\end{document} of the Laplacian on the Cayley graph satisfies λ1≥2-2cos(2π/k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _1 \\ge 2 - 2 \\cos ( 2 \\pi / k ) $$\\end{document}. We also give an explicit upper bound on the diameter of the Cayley graph of G with respect to S of the form O(k2|S|log|G|)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O (k^2 \\vert S \\vert \\log \\vert G \\vert )$$\\end{document}. Related explicit bounds for the Cheeger constant and Kazhdan constant of the pair (G, S) are also obtained.

Remarks on the continuity of convex functions in geodesic spaces

We explore the behavior of convex functions defined on locally polyhedral type spaces. As a relevant example of such spaces we show that locally compact $\mathbb{R}$-trees with the geodesic extension property around any point are locally polyhedral spaces.

Enriched multi-valued nonexpansive mappings in geodesic spaces

Enriched multi-valued nonexpansive mappings in geodesic spaces

Partially Nonexpansive Mappings in Geodesic Spaces

Partially Nonexpansive Mappings in Geodesic Spaces

Geometric statistics with subspace structure preservation for SPD matrices

We present a geometric framework for the processing of SPD-valued data that preserves subspace structures and is based on the efficient computation of extreme generalized eigenvalues. This is achieved through the use of the Thompson geometry of the semidefinite cone. We explore a particular geodesic space structure in detail and establish several properties associated with it. Finally, we review a novel inductive mean of SPD matrices based on this geometry.

On the space of null geodesics of a spacetime: the compact case, Engel geometry and retrievability

We compute the contact manifold of null geodesics of the family of spacetimes left{ left( mathbb {S}^2times mathbb {S}^1, g_circ -frac{d^2}{c^2}dt^2right) right} _{d,cin mathbb {N}^+text { coprime}}, with g_circ the round metric on mathbb {S}^2 and t the mathbb {S}^1-coordinate. We find that these are the lens spaces L(2c, 1) together with the pushforward of the canonical contact structure on STmathbb {S}^2cong L(2,1) under the natural projection L(2,1)rightarrow L(2c,1). We extend this computation to Ztimes mathbb {S}^1 for Z a Zoll manifold. On the other hand, motivated by these examples, we show how Engel geometry can be used to describe the manifold of null geodesics of a certain class of three-dimensional spacetimes, by considering the Cartan deprolongation of their Lorentz prolongation. We characterize the three-dimensional contact manifolds that are contactomorphic to the space of null geodesics of a spacetime. The characterization consists in the existence of an overlying Engel manifold with a certain foliation and, in this case, we also retrieve the spacetime.

Noncommutative spacetimes versus noncommutative spaces of geodesics

The aim of this contribution is twofold. First, we show that when two (or more) different quantum groups share the same noncommutative spacetime, such an ‘ambiguity’ can be resolved by considering together their corresponding noncommutative spaces of geodesics. In any case, the latter play a mathematical/physical role by themselves and, in some cases, they can be interpreted as deformed phase spaces. Second, we explicitly show that noncommutative spacetimes can be reproduced from ‘extended’ noncommutative spaces of geodesics which are those enlarged by the time translation generator. These general ideas are described in detail for the κ-Poincaré and κ-Galilei algebras.

Fixed point theorems for enriched Kannan mappings in CAT(0) spaces

We present enriched Kannan and enriched Bianchini mappings in the framework of unique geodesic spaces. For such mappings, we establish the existence and uniqueness of a fixed point in the setting of CAT(0) spaces and show that an appropriate Krasnoselskij scheme converges with certain rate to the fixed point. We proved some inclusion relations between enriched Kannan mapping and some applicable mappings such as strongly demicontractive mapping. Finally, we give an example in a nonlinear CAT(0) space and perform numerical experiments to support the theoretical results.

Some fixed-point theorems of convex orbital (alpha, beta )-contraction mappings in geodesic spaces

The aim of this paper is to broaden the applicability of convex orbital (alpha, beta )-contraction mappings to geodesic spaces. This class of mappings is a natural extension of iterated contraction mappings. The paper derives fixed-point theorems both with and without assuming continuity. Furthermore, the paper investigates monotone convex orbital (alpha, beta )-contraction mappings and establishes a fixed-point theorem for this class of mappings.

Convex (α, β)-Generalized Contraction and Its Applications in Matrix Equations

This paper investigates the existence and convergence of solutions for linear and nonlinear matrix equations. This study explores the potential of convex (α,β)-generalized contraction mappings in geodesic spaces, ensuring the existence of solutions for both linear and nonlinear matrix equations. This paper extends the concept to partially ordered geodesic spaces and establishes new existence and convergence results. Illustrative examples are provided to demonstrate the practical relevance of the findings. Overall, this research contributes a novel approach to the field of matrix equations.

Tree approximation in quasi-trees

n this paper, we investigate the geometric properties of quasi-trees and prove some equivalent criteria. We give a general construction of a tree that approximates the ends of a geodesic space, and use this to prove that every quasi-tree is (1,C) -quasi-isometric to a simplicial tree. As a consequence, we show that Gromov's tree approximation lemma for hyperbolic spaces (1987) can be improved in the case of quasi-trees to give a uniform approximation for any set of points, independent of cardinality. From this we show that having uniform tree approximation for finite subsets is equivalent to being able to uniformly approximate the entire space by a tree. As another consequence, we note that the boundary of a quasi-tree is isometric to the boundary of its approximating tree under a certain choice of visual metric, and that this gives a natural extension of the standard metric on the boundary of a tree.

Sufficient conditions for perturbations to define the resolvent of the equilibrium problem on complete geodesic spaces

In this paper, we consider sufficient conditions for perturbations to define a resolvent of the equilibrium problem on complete geodesic spaces with curvature bounded above.

IsoGloVe: A New Count-based Graph Embedding Method based on Geodesic Distance

Graph embedding techniques have gained increasing attention for their ability to encode the complex structural information of networks into low-dimensional vectors. Existing graph embedding methods have achieved considerable success in various applications. However, these methods have limitations in capturing global graph topology information and fail to provide insights into the underlying mechanisms of network function. In this paper, we propose IsoGloVe, a count-based method that encodes graph topology into vectors using the co-occurrence statistics of fixed-size routes in random walks. IsoGloVe calculates the final embeddings based on the geodesic distances of the node’s neighbors on a manifold. This representation in geodesic space allows for the analysis of node interactions and contributes to a better understanding of complex network structure and function. The performance of IsoGloVe is evaluated on various protein-protein interactions (PPI) using graph reconstruction, node classification, and visualization. The findings reveal that IsoGloVe surpasses other comparable methods with a 30% increase in MAP for graph reconstruction and a 25% increase in model scores for node classification in the Yeast PPI network. In addition, IsoGloVe demonstrated a 6.9% increase in MAP for graph reconstruction on the Human PPI network.

Theorems of Wolff–Denjoy type for semigroups of nonexpansive mappings in geodesic spaces

AbstractWe show that if is a one‐parameter continuous semigroup of nonexpansive mappings acting on a complete locally compact geodesic space that satisfies some geometric properties, then there exists such that S converge uniformly on bounded sets of Y to ξ. In particular, our result applies to strictly convex bounded domains in or with respect to a large class of metrics including Hilbert's and Kobayashi's metrics.

Infinite Product and Its Convergence in CAT(1) Spaces

In this paper, we study the convergence of infinite product of strongly quasi-nonexpansive mappings on geodesic spaces with curvature bounded above by one. Our main applications behind this study are to solve convex feasibility by alternating projections, and to solve minimizers of convex functions and common minimizers of several objective functions. To prove our main results, we introduce a new concept of orbital Δ-demiclosed mappings which covers finite products of strongly quasi-nonexpansive, Δ-demiclosed mappings, and hence is applicable to the convergence of infinite products.