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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What is the arc length formula in a metric space?
Let $f(x):[a,b]\longrightarrow \mathbb{R}^n$ be injective and continuously differentiable curve. Then the arc length is given by
$$\int_a^b |f'(t)|dt$$.
What will be the arc length formula if $\mathbb ...
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Given this definition of geodesic, does it have constant speed ? $f(r,t)=f(r,s)+f(s,t)$
Let $(X,d)$ be a metric space, and $x,y\in X$. In Probability Measures on Metric Spaces of Nonpositive Curvature, Sturm defines a geodesic joining $x$ and $y$ as some continuous path $\gamma :[a,b]\to ...
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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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If $f$ is differentiable at $x_0$, then the zoom converges to the straight line through the origin with slope $f'(x_0)$
A "zoom" on the graph of $y=f(x)$ near $(x_0,y_0)$ (with $y_0 = f(x_0))$ with magnification factor $M$ (the same in both $x$ and $y$ directions) is the graph of the function defined by $f(...
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Hyperbolic vs Euclidean balls
I'm trying to prove that, in the Poincaré half-space of dimension 2, a hyperbolic ball with center $P:=(x,y)$ and radius $r$ is exactly, as a set of points, a euclidean ball with center $P_1:=(x,y\...
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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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A $1$-dimensional subset like a closed curve in unit sphere connecting eight points has at least $4\pi$ length.
Consider a parallelepiped $P$ in $\mathbb{R}^3$ whose vertices are $a_i,\ 1\leq i \leq 8$. If $X$ is an interior point in convex hull of vertices, then define unit vectors $$ b_i = \frac{a_i-X}{|a_i-X|...
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Length of a geodesic in a Euclidean cone
I am studying the paper by Ohta "Products, cones and suspensions of spaces with the measure contraction property" and in the proof of Theorem 4.2 there is something I do not get.
For the ...
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Area function is continuous on a set of compact sets in $[0,1]^2$
Consider $X=[0,1]^2\subset \mathbb{R}^2$. If $H_X$ is a set of all compact sets in $X$, then we can define a metric $d$ on $H_X$ i.e. Hausdorff metric $d$ :
For $A,\ B\in H_X$, then $d(A,B)=R$ iff ...
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Example of an infinite-dimensional geodesic NPC Space
I just started reading Ballmann's book on non-positive curvature spaces. In it most, non-linear, examples of NPC spaces are negatively curved manifolds or specific graphs/discrete metric spaces, or ...
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Show that there is no distance preserving map between Mahattan norm and sup norm
Note that there is a distance preserving map between Mahattan norm and sup norm : If $A : (\mathbb{R}^2,\|\ \|_1) \rightarrow (\mathbb{R}^2,\|\ \|_\infty)$ is a linear map by $$ A(e_1)=e_1+e_2 , \ A(...
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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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A question about second fundamental form of Riemannian isometric embedding
I moved the question from mathoverflow to here.
I have got a question unsolved for some time. I do not know whether it is trivial or not:
The metric at point p is second-order flat, i.e. $d_p \phi(-,v)...
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Why a retraction from a building to an apartment is not isometric?
Let $X$ be an affine building and $\mathcal{A}$ a system of apartment. For any apartment $A\in \mathcal{A}$ and a chamber $C$ in $A$, let us consider the retraction $\rho=\rho_{A,C}\colon X\...
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Question about example of CAT($k$) space
Good time of day. I have the following question. It's written in wiki in examples of CAT($k$) spaces (https://en.wikipedia.org/wiki/CAT(k)_space) that the closed subspace $X$ of $\mathbb E^3$ (where $\...
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Length formula for Lipschitz curve, i.e. ${\rm Length}\ \gamma = \int_a^b|\gamma '(t)| dt $
I want to prove the following problem. But I think I can only
complete the half : We need to prove the part ${\rm Length}\ \gamma
\geq \int_a^b|\gamma '(t)| dt $
Problem : If $\gamma :[a,b]\rightarrow ...
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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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Two points inside a triangle x distance away from each other, at least one of it must be x distance away from one of the vertices of a triangle.
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)\...
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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). ...
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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 ...
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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($\gamma_1$)=...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $...
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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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A set of points is contained in a sphere $S$. When is $S$ also the circumsphere?
Given points $p_1,\ldots,p_n\in\Bbb R^d$ so that all of them are contained in a common sphere $S\subset\Bbb R^d$ (by which I mean the usual $(d-1)$-dimensional sub-manifold of $\Bbb R^d$). Note that $...
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Why aren't derivatives defined on metric spaces?
I'm studying the book by Ambrosio on gradient flows in metric spaces. It's stated that the usual notion of gradient flow, $$ x'(t) = -\nabla_x f(x),$$ is not defined on metric spaces because we don't ...
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Does this property of projection on metric space hold or not?
Let $(\mathcal{X}, d)$ be a complete metric space.
For any subset set $C$ of $\mathcal{X}$, we say that $C$ is convex if for any two points $x, y, \in C$, there is a geodesic $\gamma (t)$ from $x$ to $...
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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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Distance between geodesic rays
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 ...
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