# Questions tagged [metric-geometry]

The study of geometry of manifolds without appealing to differential calculus. It includes studies of length spaces, Alexandrov spaces, and CAT(k) spaces. The techniques are often applicable to Riemannian/Finsler geometry (where differential calculus is used) and geometric group theory. For questions about plain-old metric spaces, please use (metric-spaces) instead.

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### Three annulus intersection problem

Recently I faced a problem about intersection of three annuli. Imagine having three annuli same dimensions and you put them next to each other into triangular shape like putting together three circles....
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### A directionally restricted version of Bishop-Gromov inequality

Recently, I'm reading the paper "On the structure of spaces of Ricci curvature bounded below I" by J.Cheeger& T.Colding, I'm confused with the formula (2.20): Suppose $(M,p)$ is a ...
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### Whether every sufficiently good curve is almost tangent to a geodesic curve in a geodesic metric space?

Let $(X,d)$ be a strictly intrinsic metric space, that is, any pair of points in $X$ can be joined by a (not necessarily unique) rectifiable curve with the length being equal to the distance between ...
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### Extend an arbitrary multidimensional cube in such a way it stays a cube

I am searching for an equation to extend a arbitrary multidimensional cuboid in such a way it stays a cuboid. Using equations like e.g. : $f + b^2 = c^2$, $f + b^3 = c^3$, ... $f + b^n = c^n$. I am ...
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### Area function is continuous on a set of compact sets in $[0,1]^2$

Consider $X=[0,1]^2\subset \mathbb{R}^2$. If $H_X$ is a set of all compact sets in $X$, then we can define a metric $d$ on $H_X$ i.e. Hausdorff metric $d$ : For $A,\ B\in H_X$, then $d(A,B)=R$ iff ...
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### Example of an infinite-dimensional geodesic NPC Space

I just started reading Ballmann's book on non-positive curvature spaces. In it most, non-linear, examples of NPC spaces are negatively curved manifolds or specific graphs/discrete metric spaces, or ...
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### Do there exist theorems of de Sitter geometry, just as there are theorems of Minkowski and anti-de Sitter geometry?

There are theorems of Euclidean, hyperbolic, elliptic and Minkowski geometry. I'm wondering about planar de Sitter geometry. Regarding planar anti-de Sitter geometry, based on my understanding of ...
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### When does the union of all geodesics equal the metric interval?

Definitions: Throughout let $(M,d)$ be a geodesic metric space [cf. p. 104 of EoD]* with $d$ the (strictly) intrinsic metric, i.e. for all $x,y \in M$, $d(x,y)$ equals the length of any minimizing ...
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### Spectral radius of a finitely generated group

Let $G$ be a finitely generated group and $\Gamma$ be its Cayley graph with the usual word metric. Let $\mu$ be a non-degenerate measure on $G$, and define $p(x, y) = \mu(x^{-1} y)$. As is well-known, ...
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### Algebraic Characterisation of the End Space of a proper geodesic space in terms of non-continuous functions

Let $X$ be a rimcompact Tychonoff space, that is, a completely regular Hausdorff space with a base of open sets with compact boundaries. It is well known that $X$ has a maximal compactification with a ...
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### What is the degeneracy of the "Coxeter plane" for the E8 lattice?

The 240 minimal vectors (roots) of the E8 lattice, projected onto "the" Coxeter plane, are shown here: https://en.wikipedia.org/wiki/E8_(mathematics)#/media/File:E8Petrie.svg and discussed ...
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Given a triangle $\triangle$ ABC. Suppose there are two points inside the triangle, and they are $p$ and $q$. Let $d(p,q)=x$, here d represents distance. How to prove this: if $\min\{d(A,p), d(C,q)\... 1 vote 0 answers 42 views ### How to show the limit space is totally bounded in the proof of completeness of Gromov-Hausdorff metric space? The Gromov-Hausdorff metric space$(\mathcal{M},d_{GH})$is complete. I'm currently following the proof of this fact given in Petersen's Riemannian Geometry (3rd Edition) (see Proposition 11.1.8). ... • 1,115 1 vote 0 answers 119 views ### Is my proof that a constant speed geodesic is a geodesic correct? Given a metric space$(X,d)$the following is a definition of a geodesic from the book of Santambrogio. I want to show that a constant speed geodesic is a geodesic : is the following enough? Note ... • 41 0 votes 1 answer 66 views ### Asymptotic bi-infinite geodesics in CAT($-1$) space coincides Suppose$\gamma_1,\gamma_2:(-\infty,\infty)\to (X,d)$be two bi-infinite geodesic such that$d(\gamma_1(t),\gamma_2(t))<K$for all$t$. Here$(X,d)$is a CAT$(-1)$space. Then Image($\gamma_1$)=... • 2,018 2 votes 1 answer 138 views ### Existence of a geodesic with endpoints at infinity I want to prove the existence of a geodesic$\gamma$with end points$\xi$and$\xi'$at infinity in a proper CAT(-1) space. A hint in Elements of Asymptotic Geometry is to first notice that each ... • 23 3 votes 1 answer 214 views ### Derivative of distance along a smooth curve I am struggling to solve the following problem from 'Introduction to Riemannian Manifolds' by John M. Lee$(M,g)$be a connected Riemannian manifold.$\gamma:(-\epsilon,\epsilon)\to M$be a smooth ... • 2,018 1 vote 0 answers 72 views ### Quasi-isometry between an once-punctured torus and the Loch Ness Monster under suitable metrics I was wondering if there are quasi-isometric (complete) metrics on the once-punctured torus and the Loch Ness Monster. By an once-punctured torus I mean the 1-genus surface with one puncture and by ... 1 vote 0 answers 73 views ### Floyd Compactification Suppose$G$is a finitely generated group,$f: \mathbb{N} \longrightarrow \mathbb{R_+}$is a function satisfying$1<\frac{f(n)}{f(n+1)}<\lambda$and$\sum f(n)<\infty$, which we called a ... • 11 2 votes 1 answer 142 views ### Gromov product on the boundary I was reading the book Elements of Asymptotic Geometry by Sergei Buyalo and Viktor Schroeder. There in the second chapter Gromov boundary is defined as the equivalence classes of sequences (in the ... • 2,018 1 vote 2 answers 98 views ### Borel quasi-isometry between proper geodesic spaces Let$(S,d_{1})$and$(S',d_{2})$be two proper geodesic metric spaces. If there exists a quasi-isometric embedding$f: S\rightarrow S'$, does there exist a$\textbf{Borel}$quasi-isometric embedding$...
Let $(S,d_{1})$ and $(S',d_{2})$ be two proper geodesic metric spaces. If there exists a quasi-isometric embedding $f\colon S\rightarrow S'$, does there exist a $\textbf{continuous}$ quasi-isometric ...