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 ...

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5
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82 views

Why is the Barycenter operation in Hadamard spaces Lipschitz continuous?

I am looking into exercise 9.2.22 of "A course in metric geometry" by Burago-Burago-Ivanov. For a Hadamard space $H$ (a complete simply connected metric space of nonpositive curvature in the ...
0
votes
1answer
43 views

Convex subsets of pinched Hadamard manifolds

Let $X$ be a pinched Hadamard manifold (in my particular case, $X=\mathbb H^n$ is the $n$-dim. hyperbolic space) and $N$ be a closed (edit : open) convex subset of $X$. Is it true that $N$ is also a ...
0
votes
1answer
60 views

A notion of nonpositive curvature for general metric spaces

The proof of the following result should be done by using the second variation formula of geodesics but I do not know how to start or what is the main idea of the proof. (Lemma 3.7 in the paper: A ...
0
votes
2answers
45 views

Distance inequality

Consider $H=\{ (x,y) \in {\bf R}^2\mid x$ or $y$ is an integer $\}$ If $d$ is canonical distance in ${\bf R}^2$, show that if $d(x):=d(x,H)$, (1) $$ d(x) - d(y) \leq d(x,y) $$ if $x,\ y$ are in same ...
3
votes
0answers
22 views

Christoffel Symbols in terms of the Log Function

Since the Riemmanian Log function expresses the Manifold structure in terms of $\mathbb{R}^d$ locally, then I was wondering: Can we express the Christoffel Symbols explicitly in terms of the Log ...
0
votes
0answers
23 views

Extension of divergence free vector field as a divergence free vector field.

Let $M$ be a compact smooth Riemannian manifold of dimension $n$. Assume that $M$ is isometrically embedded in $\mathbb{R}^m$ for some sufficiently large $m$ via the map $\iota$. Let $X:M\to TM$ be ...
4
votes
1answer
45 views

Riemmanian Distance is always greater?

Setup: Suppose $M$ is a $C^k$-manifold embedded into some Hilbert space $H$ and $g$ is the induced Riemmanian metric thereon (induced by restricting the inner-product $\langle,\rangle_H $ in $H$ to ...
6
votes
1answer
94 views

Lower semi-continuity of one dimensional Hausdorff measure under Hausdorff convergence

Let $\mathcal H^1$ be the one-dimensional Hausdorff measure on $ \mathbb R^n$, and let $d_H$ be the Hausdorff metric on compact subsets of $\mathbb R^n$. If $K_n$ is connected for all $n \in \mathbb ...
5
votes
0answers
24 views

Global Chart implies no cut locus?

If a manifold $M$ admits a global chart, does this imply that there exists a point $p\in M$ such that $Cut_p=\emptyset$? Recall: Definition of $Cut_p$: Let $\mathfrak{C}_p$ be defined as the set ...
2
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0answers
13 views

Quantifying the angle metric on the Grassmannian in terms of the norm on the exterior power

Let $V$ be a finite-dimensional Hilbert space and $G_k(V)$ the Grassmannian of $k$-dimensional subspaces of $V$. The $k$th exterior power $\bigwedge^k(V)$ can be equipped with a scalar product by ...
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0answers
40 views

What simple topological properties of conic sections can be explored?

In the framework of my science fair project I am working on conic sections in different metric spaces. What simple topological properties/operations and so can I explore on them? Edit: To clarify, ...
2
votes
1answer
66 views

Alternative Geometries

In our world, the distance between two points (in 2d) is defined as $\sqrt{(\Delta x)^2 + (\Delta y)^2}$. Suppose that in an alternative geometry, it was defined as $\sqrt[p]{|\Delta x|^p + |\Delta ...
0
votes
2answers
38 views

Hausdorff dimension of homeomorphic compact metric spaces

Are there examples of homeomorphic compact metric spaces of different Hausdorff dimension? If yes, are there sufficient conditions on the spaces which would imply the equality of Hausdorff ...
2
votes
1answer
55 views

Centralizers in mapping class groups

According to Nielsen-Thurston classification, given a closed surface $S$, the elements of the mapping class group $\mathrm{MCG}(S)$ lie in three categories: periodic, reducible and pseudo-Anosov. ...
0
votes
1answer
35 views

A complete metric space with approximate midpoints is intrinsic

Let $(X,d)$ be a metric space. For $x,y\in X$, define $A_{xy}$ the set of curves (the domain is supposed to be $[0,1]$) joining $x$ with $y$. For $\sigma\in A_{xy}$, define its length as ...
2
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0answers
53 views

How can the notion of “two curves just touching” (vs. “two curves intersecting”) be expressed for a given metric space?

A popular introductory description of a "tangent (in geometry)" is presented as "the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the ...
1
vote
1answer
60 views

Given a spacetime in terms of values of Lorentzian distance, how to determine whether it is flat?

Yesterday I learned that geometric relations between events can be characterized generally (and up to a common non-zero factor) in terms of their pairwise "Lorentzian distance, $d_{\ell}$", which ...
3
votes
0answers
89 views

Taylor expansion of the square of the distance function

Give a smooth Riemannian manifold $(M,g)$, (i) how can one compute Taylor expansion of the square of the Riemannian distance function $d^2(x,x_0)$ at $(x',x_0)$? I've tried to use $dist=\int ...
0
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0answers
37 views

Connections between Level Sets and convex/concave Functions

I am interested in the connection between level sets and convex/concave functions on Manifolds. For example if a Function is convex (concave) all sublevel (superlevel) sets are convex. Is the ...
0
votes
1answer
48 views

Concavity of distance function in $\mathbb{R}^n$ or determinant of $(x^T \cdot x)$

I would like to compute the concavity of the distance function in $\mathbb{R}^n$. Let $ f(x) =- \Vert x \Vert $ in $\mathbb{R}^n$. Then $\nabla_xf=- \frac{x}{\Vert x \Vert}$. And ...
3
votes
1answer
50 views

Are Homogenous countable complete metric spaces always discrete?

Let $M$ be a countable complete metric space such that the group of isometries of $M$, $Iso(M)$ acts transitively on the points in $M$. Does it follow that the topology induced by the metric is the ...
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vote
0answers
76 views

How to find the “area” of a box (hyperrectangle) projected onto a ball (hypersphere)?

What is the "area" of the general $n-1$ - box (hyperrectangle) projected onto the surface of the general $n$ - ball (hypersphere)? I'm just curious.
4
votes
1answer
134 views

Moving circular disk between two parallel sinusoidal curves

Find the largest radius of the circle that can be "rolled" between the curves $y = sin(x)$ and $y = sin(x)+1$. After two weeks of research, I finally give up.
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0answers
41 views

Prerequsites for working through the 2nd half of Gradient flows in metric spaces and in the spaces of probability measures

I apologize in advance if this question is too general, that is, not a request for a specific reference, but more of a request for a road map, perhaps from someone that knows the material and, in ...
0
votes
1answer
142 views

upper bound and a lower bound on the number of points that are uniformly distributed on a surface

Can I calculate an upper bound and a lower bound (or max or min) on the number of points that are uniformly distributed on a surface, knowing the area of the surface ? More precisely, I have a sector ...
1
vote
1answer
37 views

Given a non-ideal hyperbolic triangle and the Euclidean comparison triangle with equal side lengths, are the interiors of the two bi-Lipschitz?

Fix three finite real numbers $p,q,r > 0$. Up to isometry, there is a unique 2-simplex $\Delta$ in the Euclidean plane bounded by a geodesic triangle with these three reals $p,q,r$ as side-lengths. ...
0
votes
0answers
62 views

On the centroid of a triangle

There's three different ways to see a triangle in the Euclidean plane: as three non-collinear points, say $A$, $B$, $C$; as the line segments connecting the three points, that we can parametrize as a ...
1
vote
1answer
35 views

CAT(K) Finsler manifolds.

I was wondering if the following is true (and common knowledge): Let $(M,F)$ be a Finsler manifold. Let d be the induced distance by the norm in the usual sense. That is, $d(x,y)=\inf${lenghts of ...
1
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0answers
61 views

A measure on the space of probability measures

I've been reading about optimal transport and it's connections to geometry. At some point one has to study a bit of the structure of the space of probability measures, $\mathcal{P}(X)$, (over a metric ...
1
vote
1answer
36 views

Invariance of ball size under translation on the pseudosphere

On the pseudosphere, do the volume of metric balls and area of metric spheres change with the centerpoint? (I'm asking because they don't on the sphere.)
4
votes
1answer
123 views

How to build the smallest regular n-sided polygon that covers an (n-1)-sided polygon?

I want to build a figure that contains seven regular polygons, from a triangle up to a nonagon, where each n-sided polygon covers, with the minimal area possible, the n-1 sided one. An added ...
2
votes
1answer
97 views

why are CAT(0) spaces contractible?

In the book of Bridson and Haefliger it is said that 'it follows easily' from what they proved before. Does anyone know of a rigorous proof that CAT(0) spaces are contractible?
3
votes
1answer
216 views

general formula for volume of a simplex?

I am looking for a general formula to calculate the volume of a euclidean simplex in any number of dimensions. On Wikipedia I found that a formula similar to Heron's formula can be applied to ...
4
votes
2answers
79 views

Ultraproduct of a metric space

I am currently trying to understand "Curvature bounded below: a definition a la Berg--Nikolaev" by Nina Lebedeva and Anton Petrunin. They start with a complete, intrinsic metric and space $X$ and say ...
2
votes
1answer
70 views

Closure of a region and shortest path

Let $\Omega$ be a region in $\mathbb{R}^2$ and $\overline{\Omega}$ be closure of $\Omega$. Is it true that between every two points $x,y \in \overline{\Omega}$ exists shortest path (lenth of path ...
0
votes
1answer
248 views

How to calculate the center of a regular polygon?

What is the formula for the center of an n-edge regular polygon that has the given segment as its edge? So, given a segment AB, ...
2
votes
1answer
61 views

Linear bound on angles in an euclidean triangle.

I am trying to understand a proof in the book of Burago "A Course in metric geometry" (Lemma 10.8.13 page 383). I have difficulties with a certain inequality for the angle of euclidean triangles: ...
1
vote
1answer
77 views

Existence of CAT(0)-metrics

Given a metrisable, contractible topological space. Under which conditions exists a CAT(0)-metric compatible with the given topology?
1
vote
2answers
123 views

How to show the Poincaré disk is hyperbolic for some $\delta$

I am trying to prove that the Poincaré disk, $\mathbb{D}$, is $\delta$-hyperbolic with respect to the slim triangle definition for hyperbolicity. I have been stuck for a while on where to begin, ...
7
votes
0answers
163 views

Gromov-Hausdorff distance between a line segment and a cylinder

I want to prove the following statement, where $d_{GH}$ denotes the Gromov-Hausdorff distance: For $X= S^1 \times [0,1]$ and $Y= [0,1]$ with the intrinsic metrics one has $d_{GH}(X,Y) = ...
1
vote
1answer
180 views

Lipschitz distance

The Lipschitz distance between two metric spaces is defined by $$d_{\mathcal L}(X, Y) = \inf_f \log(\{\max\{\text{dil}(f), \text{dil}(f^{-1})\})$$ where $$\text{dil}(f) = \sup_{x_1, x_2 \in X} ...
1
vote
1answer
52 views

Strainers in Alexandrov spaces

I am reading the section on Strainers in Burago, Burago and Ivanov's book "A Course in Metric Geometry". I have been struggling with the proofs of some of the lemmas. On Lemma 10. 8. 13, the authors ...
1
vote
1answer
47 views

Maximal Hausdorff dimension of the inverse image of a point

Let $f \colon [0,1]^2 \to \mathbb{R}$ be an arbitrary continuous function. I was wondering the following: Does there exist a point $a \in \mathbb{R}$ such that $f^{-1} \{a\}$ has Hausdorff dimension ...
2
votes
0answers
40 views

Squaring the plane with consecutive integer squares. And a related arrangement. [duplicate]

Q1. I was fiddling around with squaring-the-square type algebraic maths, and came up with a family of arrangements of $n^2 $squares, with sides $1,2\ldots n^2$ (n odd). Which seems like it would work ...
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vote
1answer
85 views

Equivalent Definition of a Hermitian Metric on an Almost Complex Manifold?

For an almost-complex manifold $M$ with almost-complex structure $J$, we say that a metric $h:T_p(M;{\mathbb R}) \times T_p(M;{\mathbb R}) \to {\mathbb R}$ is Hermitian if it holds that $$ ...
13
votes
1answer
347 views

Gromov-Hausdorff convergence to a circle

I am working on the book A course in metric geometry written by D. Burago, Y. Burago and S. Ivanov, and more precisely on exercice 7.5.9: Exercice: Let $\{X_n\}$ be a sequence of compact length ...
3
votes
1answer
91 views

How much close should two spaces be in the Gromov-Hausdorff distance to be homeomorphic?

Here I am considering the Gromov-Hausdorff convergence for metric spaces. I know two compact metric spaces are isometric if and only if $d_{GH}(X,Y)=0$, where $d_{GH}$ denotes the Gromov-Hausdorff ...
0
votes
0answers
223 views

Isometry groups acting transitively

Let $X$ be a metric space and $G$ be its group of isometries. 1) Is it true that $G$ acts on $X$ transitively? If so, where can I find a proof? If not, how can one characterize those $X$ for which ...
1
vote
0answers
97 views

vertex representation and half-space representation of a polytope

i'm not a mathematician. So, my question may be stupid. Hope that it won't make you angry. Is there any mathematical relationship between the vertex representation and the half-space representation ...
1
vote
1answer
95 views

Sylvester-Gallai Theorem

How is this theorem used in applications? I've been searching for it on the web but can't seem to find. Only to "correct codes". Can someone please give a few simple examples? /lost student