# 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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### sufficient condition for a finite dimensional normed real vector space to be a hilbert space [closed]

This comes from Exercise 1.2.24 from the book: A course in metric geometry by Dmitri Burago, Et al. Let $V$ be a finite dimensional normed real vector space, if for any $x,y\in V$ with the same norm, ...
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### Book recommendations for finite distance and angle solving-oriented, plane and/or solid geometry

I am looking for books that: having same spirits to many books that commonly known for their comprehensive treatment of Euclidean geometry (e.g. this). I am really aware to kind of those books ...
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### Question on the Palatini identity for the metric variation of the Riemann tensor

Let $({X},\mathcal{O},\mathcal{A},{g},\nabla)$ be a smooth ${n}$-dimensional manifold with Riemannian metric ${g}$ and the Levi-Civita connection ${\nabla}$. We now define an action functional that is ...
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### To define an ellipse in terms of a pair of points that lie on its minor axis

Just as an ellipse can be defined in terms of the sum of distances from its foci, that are both points on the major axis, can any ellipse be defined with reference to a pair of suitably chosen points ...
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### Bi-infinite geodesic in geodesic Gromov-Hyperbolic spaces

For a geodesic Gromov Hyperbolic metric space X is it true that there exists $C>0$ such that any two bi-infinite geodesic with same end points at boundary stays within $C$-neighbouhood of each ...
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### Bound of Hessian in Wasserstein Gradient Flow

Consider two measures $\mu, \nu \in \mathcal{P}^r_2(\Theta)$ with densities and finite second-order moments on a set $\Theta\subset\mathbb{R}^d$. Let $T_\mu^\nu$ be the optimal map from $\mu$ to $\nu$,...
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### Bijective and distance preserving map between finite dimensional normed vector space is not a linear map

If $f$ is a distance preserving map from Euclidean space to itself, then I can show that it is infact a composition of linear maps and translations. If $f$ is a bijective and distance preserving ...
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### A map which is homeomorphism and isometry but not diffeomorphism

The question comes from this about metric space which is also smooth manifold. Existence of a Riemannian metric inducing a given distance. Alexandrov proved that Suppose that $(𝑀,𝑑)$ is a locally ...
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### Local Lipschitz constants and a slightly weaker concept

Let $(X, d)$ be a metric space and $f:X \rightarrow \mathbb R$ be a function such that $$\lim_{r \rightarrow 0} \sup_{0 < y < B(x; r)} \frac{|f(x) - f(y)|}{d(x, y)}$$ exists for some $x \in X$. ...
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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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### Is every coarse map between proper geodesic spaces a quasi-isometric embedding?

Let $(S,d_{1})$ and $(S',d_{2})$ be two proper geodesic metric spaces. Is every coarse map $f: S\rightarrow S'$ a quasi-isometric embedding?. Just to recall, a coarse map $f: S\rightarrow S'$ between ...
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### K-convexity, duality and in Banach Spaces

I am interested in K-convexity as defined in https://webusers.imj-prg.fr/~bernard.maurey/articles/typandco.pdf Page 24 bottom of page. I am struggling to show that K-convexity passes to the dual given ...
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### Geodesic convexity of small balls in Alexandrov CBB spaces

If $X$ is a Riemannian manifold, it is known that, for $p\in X$, there is some $\epsilon>0$ such that $B(p,\epsilon)$ is geodesically convex. Geodesically convex means there is a minimizing ...
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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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### Continuous quasi-isometry between Riemannian manifolds

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 ...
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### Total Curvature for a curve in a metric space

Is there a theorem involving total curvature and some notion of index number for a curve in a metric space, as there is for planar curves? (i.e. total curvature is an integer multiple of $2\pi$.) I'm ...
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### Does the Besicovitch Covering Property hold on finitely generated groups of polynomial growth when equipped with the word length metric?

Let $G$ be a finitely generated group with symmetric generating set $S$. Then $S$ induces a distance $d$ on $G$ by letting $d(a,b) =$ the minimum $n$ such that there are generators $s_1,...,s_n$ with ...
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### Question on the distortion of a metric embedding and Lipschitz maps

This is a bit of mild confusion off of Matoušek's lecture notes on metric embeddings (available at https://kam.mff.cuni.cz/~matousek/ba-a4.pdf). An injection between metric spaces $f : X \rightarrow Y$...
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### If an isometry $f$ of a length space $X$ fixes a ball $B(x,r)$ for some $x$ and $r>0$, is it true that $f$ is the identity map?

Let $(X,d,L)$ be a length space, that is, $d = \inf L$, where the inf is taken over all curves with fixed endpoints. In $\mathbb{R}^2$ we have a results that says: if an isometry fixes 3 points, then ...
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### Distance between geodesic rays 2

This is a follow up to my previous question which turned out to be wrong. Now my question instead is this: Given a $\delta$-hyperbolic space $(X,d)$ (with Rips triangle condition: a point on any side ...
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I am trying to prove the following but so far did not succeed. Given a $\delta$-hyperbolic space $(X,d)$ (with Rips triangle condition: a point on any side is contained in the $\delta$-neighbourhood ...