Related papers: Gromov-Hausdorff class: its completeness and cloud…

In spacetime physics, we frequently need to consider a set of all spaces (`universes') as a whole. In particular, the concept of `closeness' between spaces is essential. However, there has been no established mathematical theory so far…

The original proof of the Gromov's non-squeezing theorem [Gro85] is based on pseudo-holomorphic curves. The central ingredient is the compactness of the moduli space of pseudo-holomorphic spheres in the symplectic manifold…

We survey some basic results on the Gromov-Prohorov distance between metric measure spaces. (We do not claim any new results.) We give several different definitions and show the equivalence of them. We also show that convergence in the…

Here we extend the notion of target-local Gromov convergence of pseudoholomorphic curves to the case in which the target manifold is not compact, but rather is exhausted by compact neighborhoods. Under the assumption that the curves in…

A totally umbilical submanifold in pseudo-Riemannian manifolds is a fundamental notion, which is characterized by the condition that the second fundamental form is proportional to the metric. It is also a generalization of the notion of a…

We prove a compactness result for classes of actions of many small categories on quantum compact metric spaces by Lipschitz linear maps, for the topology of the covariant Gromov-Hausdorff propinquity. In particular, our result applies to…

In this paper, we first prove that any power quasi-symmetry of two metric spaces induces a rough quasi-isometry between their infinite hyperbolic cones. Second, we prove that for a complete metric space $Z$, there exists a point $\omega$ in…

We describe a natural isomorphism between the set of equivalence classes of pseudocycles and the integral homology groups of a smooth manifold. Our arguments generalize to settings well-suited for applications in enumerative algebraic…

The modular Gromov-Hausdorff propinquity is a distance on classes of modules endowed with quantum metric information, in the form of a metric form of a connection and a left Hilbert module structure. This paper proves that the family of…

Let $X$ be a finite metric space with elements $P_i$, $i=1,\dots,n$ and with distance functions $d_{ij}$. The Gromov product of the triangle with vertices $P_i$, $P_j$ and $P_k$ at the vertex $P_i$ is defined by…

We study the geometry of the space of densities $\VolM$, which is the quotient space $\Diff(M)/\Diff_\mu(M)$ of the diffeomorphism group of a compact manifold $M$ by the subgroup of volume-preserving diffemorphisms, endowed with a…

It is shown that, in the Gromov space of isometry classes of pointed proper metric spaces, the equivalence relations defined by existence of coarse quasi-isometries or being at finite Gromov-Hausdorff distance, cannot be reduced to the…

We introduce the concept of a class of equivalence of molecular clouds represented by an abstract spherically symmetric, isotropic object. This object is described by use of abstract scales in respect to a given mass density distribution.…

Urysohn constructed a separable complete universal metric space homogeneous for all finite subspaces, which is today called the Urysohn universal metric space. Some authors have recently investigated an ultrametric analogue of this space.…

Gromov-Hausdorff distances measure shape difference between the objects representable as compact metric spaces, e.g. point clouds, manifolds, or graphs. Computing any Gromov-Hausdorff distance is equivalent to solving an NP-Hard…

The Gromov-Hausdorff distance $(d_{GH})$ proves to be a useful distance measure between shapes. In order to approximate $d_{GH}$ for compact subsets $X,Y\subset\mathbb{R}^d$, we look into its relationship with $d_{H,iso}$, the infimum…

We prove that the sequence of cones of metric measure spaces converges if the sequence of base spaces converges in Gromov's box, concentration, and weak topologies. As an application, we show that the generalized Cauchy distribution with…

We consider Gromov-Hausdorff convergence of state spaces for spectral truncations of a compact metric group $G$. We work in the context of order-unit spaces and consider orthogonal projections $P_\Lambda$ in $L^2(G)$ corresponding to finite…

The Hausdorff hyperspace of a metric space consists of all its non-empty bounded closed sets and it is equipped with the Pompeiu--Hausdorff set distance. We present a simpler novel proof that the Hausdorff hyperspace of a complete space is…

We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two metric spaces $X$ and~$Y$ can be lower bounded by the…