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....
Stefan Vujic's user avatar
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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 ...
eulershi's user avatar
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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 ...
Rafael's user avatar
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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 ...
UweJ's user avatar
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Doubt about exercise on family of circles

Consider the following family of cirles $$ x^2+y^2-2x+4y-2+k(x^2+y^2-6y+8)=0 $$ We are asked about the value of $k$ so that the circle will have the center on the straight line given by the equation $$...
Francesco Bruno's user avatar
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A property of complete metric spaces makes them length (path or inner) metric spaces, Clarification of a proof

In the book "Metric Structures for Riemannian and Non-Riemannian Spaces", by Misha Gromov, I found a proof of the following statement (of Theorem 1.8. restated here more concentrated) Let $(...
Physor's user avatar
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The meaning of the equation of the line that passes through a point and intersects with a vector

I have a question about the meaning of a system of equations represented in R3 (3 dimensions if I'm not mistaken), in which we are finding the intersection of vector u (equation 1) with the line ...
Ángel José Álvarez Pérez's user avatar
3 votes
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129 views

The length of shortest curve dividing topological sphere into two parts of same area

$(S^2,g)$ is a Riemannian manifold, where $S^2$ is homeomorphic to the 2-dimensional sphere. And the area of $(S^2,g)$ is $4\pi$. $\gamma$ is the shortest curve dividing $S^2$ into two parts of equal ...
Enhao Lan's user avatar
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Intuition for finding a circumcenter using cross products?

The popular way to determine the circumcenter of a triangle is to find the intersection of two perpendicular bisectors, but I noticed the wikipedia page for circumscribed circle also describes a ...
prideout's user avatar
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Question about geodesic

On the Wikipedia page for Geodesic, it's stated that a curve $\gamma : I → M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v \geq 0$ such that for ...
Squirrel-Power's user avatar
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Are Hyperbolic Angles Smaller than Euclidean ones in 'Congruent' Triangles?

Assume you have a triangle with vertices $A,B,C\in\mathbb H^2$ in the hyperbolic plane and the hyperbolic distances are $a=d(B,C), b=d(A,C), c=d(A,B)$. Now pick a `comparison triangle' $A',B',C'\in\...
Franz Luggin's user avatar
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Length space and continuity assumption on curves

I think the definition of length metric works without assuming curves are continuous. Let $X$ be a subset of $\mathbb{R}^n$ and $x, y\in X$. Def 1. For a function $f:[0, 1]\rightarrow X$, $L(f)$ is ...
BonBon's user avatar
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Are Riemannian manifolds isotropic if and only if they satisfy the SAS postulate? If not, is either class more general?

Question: Assume $M$ is a connected, smooth, and geodesically complete Riemannian manifold. (Hence $M$ is a complete metric space by the Hopf-Rinow theorem.) Then (assuming that $M$ satisfies the SAS ...
hasManyStupidQuestions's user avatar
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Do general Riemannian manifolds satisfy the SAS (side-angle-side) postulate?

By triangle I have in mind something where all sides must be length minimizing geodesics, i.e. geodesic segments. In particular the distances between their endpoints will be the length of the geodesic....
hasManyStupidQuestions's user avatar
3 votes
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Approximating $\varepsilon$-isometries by continuous $\varepsilon$-isometries (Burago-Burago-Ivanov Exercise 7.5.8)

Given compact metric spaces $X, Y$, a function $f \colon X \to Y$ is an $\varepsilon$-isometry if $f(X)$ is an $\varepsilon$-net in $Y$ (every point in $Y$ is within $\varepsilon$ of a point in $f(X)$...
ckefa's user avatar
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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 ...
stephan's user avatar
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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 ...
Marlou marlou's user avatar
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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 ...
làntèrn's user avatar
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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 ...
Nandakumar R's user avatar
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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 ...
Noob mathematician's user avatar
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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(...
Real Analysis Noob's user avatar
1 vote
1 answer
111 views

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\...
Lille Nordmann's user avatar
1 vote
1 answer
53 views

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|...
HK Lee's user avatar
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3 answers
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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 ...
HK Lee's user avatar
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2 votes
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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 ...
Mike_MasterinMath's user avatar
2 votes
1 answer
36 views

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(...
HK Lee's user avatar
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1 vote
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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 ...
HK Lee's user avatar
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3 votes
1 answer
142 views

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 ...
threeautumn's user avatar
1 vote
1 answer
32 views

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 $x \in X$. ...
katagiri's user avatar
1 vote
1 answer
106 views

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)...
threeautumn's user avatar
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54 views

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 $\rho=\rho_{A,C}\colon X\...
M masa's user avatar
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0 answers
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Question about example of CAT($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 $\...
Victory's user avatar
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1 vote
2 answers
197 views

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 ...
HK Lee's user avatar
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3 votes
0 answers
50 views

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 ...
wlad's user avatar
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2 votes
0 answers
58 views

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 ...
hasManyStupidQuestions's user avatar
4 votes
0 answers
57 views

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, ...
SMS's user avatar
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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 ...
Carlos Adrián's user avatar
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63 views

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 ...
Dan Haxton's user avatar
1 vote
1 answer
37 views

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)\...
maths123456's user avatar
1 vote
0 answers
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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). ...
Hopf eccentric's user avatar
1 vote
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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 ...
hopper's user avatar
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1 answer
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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($\gamma_1$)=...
Noob mathematician's user avatar
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 ...
George's user avatar
  • 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 ...
Noob mathematician's user avatar
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 ...
Carlos Adrián's user avatar
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 ...
ggt001's user avatar
  • 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 ...
Noob mathematician's user avatar
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 $...
Carlos Adrián's user avatar
2 votes
2 answers
169 views

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 ...
Carlos Adrián's user avatar

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