Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.
4
votes
1answer
79 views
Conformally flat homogeneous spaces
Let's say we have a homogeneous space $H\backslash G$.
Is it possible to tell whether this homogeneous space admits a conformally flat metric just from its group structure?
I am particularly ...
4
votes
1answer
90 views
Is a symmetric, parallel (0,2)-tensor a metric?
I'm interested in affinely connected spaces, on which a metric is not necessarily defined, i.e. $(\mathcal{M},\Gamma)$. Since (as a physicist) my goal is to consider a generalized model of gravity, I ...
-1
votes
0answers
59 views
Prove or disprove the non colinearity of three points
Prove or disprove: Let $ABCD$ be a quadrilateral in Pasch Geometry for which the following holds $AB\cap CD=\{E\}$, $AC\cap BD=\{F\}$ and $AD\cap BC=\{G\}$. Then the points $E,F,G$ are not collinear.
...
1
vote
1answer
46 views
How do I find the correct distance spacing for distributing equidistant points on a sphere of a given diameter?
How do I find the correct distance spacing for distributing equidistant points on a sphere of a given diameter?
I don't need to fill the sphere with equidistant points. I just need less than a ...
4
votes
0answers
60 views
Optimal placement of points on sphere to minimize size of projections
While studying some physical problem we stumbled across a geometrical problem, to which we could not find any solution.
Consider a unit sphere in 3D and a collection of $N$ points on its surface.
Now ...
3
votes
0answers
58 views
Reference of generalized isometries
I'm wondering if these objects have a name or are studied. No one around me knows, so I thought to ask here.
Let $\Phi:\mathbb{R}^d\rightarrow \mathbb{R}^d$ be $C^2$-diffeomoprhism, and fix $p \geq ...
5
votes
0answers
100 views
Does $G$ act 2-transitively on its Bruhat-Tits building?
Let $k$ be a finite extension of $\mathbb{Q}_p$ and let $G$ be a semisimple Lie group over $k$. We consider the action of $G$ on its Bruhat-Tits building $X$.
Question: If $x,y,x',y'$ are vertices, ...
-1
votes
0answers
45 views
n-polytope with equal area faces
For which n does there exist a convex polytope with n equal-area faces? For n=4,6,8,12,20 the answer is trivially yes (Platonic solids). I think the answer is yes and have a rough idea of the proof -...
1
vote
1answer
75 views
A question on a special “metric”
Suppose we have a function $F: [a,b]^n \to \mathcal{M}_{n \times n }(\mathbb{R})$ where $\mathcal{M}_{n \times n }(\mathbb{R})$ is the space of $n \times n$ real matrices, a compact set $B \subset \...
6
votes
1answer
199 views
What is the geometric meaning of one Riemannian metric bigger than the other one on a smooth manifold?
Gromov conjectured in 1985 and LLarull proved in 1998 that: If $g > g_0$ on the sphere, then there exists some point p on the sphere with $Sc(p) < Sc_0(p)$. Here $g, g_0$ are Riemannian metrics ...
2
votes
0answers
32 views
Rough classification of segment metrics
For a dynamical project, I need to bound uniformly a quantity defined from a metric on the segment $I=[0,1]$ over all metrics inducing the usual topology. The details do not matter too much, I wonder ...
0
votes
0answers
42 views
Does lattice mod preserve direction?
For high enough dimension $n$ there are lattices $L_n$ in $\mathbb{R}^n$ whose Voronoi partition's base regions encompass all but a negligible proportion of a $(1-\varepsilon)$-ball, and also nearly ...
-1
votes
1answer
36 views
Characterisation of a superset of the simplex
Does there exist a nice description of the following set:
\begin{equation}
A:=\left\lbrace x\in\mathbb{R}^{n}\ \colon\ 0< x_{i}-\bar{x}+\frac{1}{n}< 1\ \text{for} \ i=1,\dots,n\right\rbrace,
\...
7
votes
3answers
317 views
Sectional curvature of leaves of foliation
Given a $k$- dimensional foliation $F$ of a riemannian $n$-manifold $M$, with the property that the leaves of the foliation have constant sectional curvature $s$, for some $s$, is it true that $M$ ...
21
votes
7answers
2k views
Smooth functions on sphere
Let $u$ be a smooth function defined on the unit sphere $S^2$. Assume $u$ has two local maxima, two local minima, and two saddle points (a total of 6 critical points). Does there exist a plane $P$ ...
2
votes
2answers
77 views
Broken geodesic in Finsler polyhedral space
Here we assume that all norms has only one geodesic, i.e. locally
minimizing, between any two points.
Example : In $\mathbb{R}^2$, a line $y=kx,\ k>0$ divides
$\mathbb{R}^2$ into two regions. We ...
6
votes
0answers
96 views
Can scalar curvature and diameter control volume? Round 2
This is a follow up to a question by Yiyue Zhang. Can scalar curvature and diameter control volume?
The original question asked whether scalar curvature bounds and small diameter bounds were enough ...
14
votes
3answers
308 views
What is known about sufficient conditions for the rigidity of a convex surface?
A convex surface is a connected open subset of the boundary of a convex body in $\mathbb{R}^3$.
An "infinitesimal bending" of a convex surface $S$ is a deformation of $S$ given by a velocity field $v:...
5
votes
2answers
107 views
Function as sum of distances over a connected, compact metric space
If $X$ is a connected, compact metric space with distance function $d : X^2 \rightarrow \mathbb{R}^+$, is it true that there exists a positive real number $a$, dependent on $X$ and $d$, such that for ...
18
votes
1answer
298 views
Gluing hexagons to get a locally CAT(0) space
I believe that there are four ways to glue (all) the edges of a regular Euclidean hexagon to get a locally CAT(0) space:
The first two give the torus and the Klein bottle, respectively. What are the ...
3
votes
0answers
101 views
An new equilateral triangle related to the Morley triangle
Morley equilateral triangle is the nice theorem in Eulidean Geometry. I found an equilateral triangle and a group circle related to the Morley triangle and angle trisectors:
Let $ABC$ be a triangle ...
1
vote
0answers
59 views
Some hypersurface has a positive second fundamental form potentially
Notation : $r^2=x^2+y^2$.
Exercise : Define $$F_\sigma (x,y)= (f_\sigma
(x,y),\frac{x^2-y^2}{2},xy)$$ where $$f_\sigma (x,y)=(1- \sigma^2
r^2)(x-\sigma x^3,y-\sigma y^3)$$
Define $ G_\sigma: \mathbb{...
6
votes
0answers
92 views
Do manifolds with non-negative Ricci curvature allow bi-Lipschitz embeddings into Euclidean spaces?
QUESTION: Let $n$ be a natural number. Is it true that there exist $N(n), D(n) > 0$ such that any complete $n$-dimensional Riemannian manifold of nonnegative Ricci curvature can be embedded into $N$...
4
votes
2answers
147 views
largest diameter of intersection of two balls
Two closed balls with a common radius are positioned so that the centre of either ball is on the boundary of the other.
I am interested in the extremal diameter of their intersection, in an arbitrary ...
4
votes
0answers
124 views
Non-algebraic quasi-isometric embeddings
What are examples of finitely generated groups $\Gamma$ and $\Lambda$ such that the metric space $\Lambda$ embeds into $\Gamma$ quasi-isometrically but such that $\Lambda$ is very much not a subgroup ...
3
votes
0answers
200 views
Are these points known? [closed]
Let $ABC$ be a triangle and $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $A'$, $B'$, $C'$ respectively.
From my construction by GeoGebra, I found two special points as ...
1
vote
0answers
76 views
DIstance on a Riemannian manifold [closed]
Given a Riemannian manifold $(M,g)$ is it possible to calculate the distance between two points on this manifold. Is it possible the inverse? That means: given a formula of the distance, for example:
...
6
votes
0answers
88 views
Geometric mean of three or more positive definite matrices
The geometric mean of two positive definite (Hermitian) matrices of same size is defined by $$A\natural B := A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$$equivalently,
$$A\natural B =(BA^{-1})^{1/2}A=A(A^...
4
votes
0answers
146 views
Uniqueness of the boundary of a hierarchically hyperbolic group
Hierarchically hyperbolic groups and spaces (HHG and HHS for short) were defined by Behrstock, Hagen and Sisto (see here and here). Examples include mapping class groups, Right angled Artin groups, ...
13
votes
0answers
213 views
Why are the medians of a triangle concurrent? In absolute geometry
This fact holds true in absolute geometry, and I would like to see an elementary synthetic proof not using the classification of absolute planes (Euclidean and hyperbolic planes) and specific models. ...
3
votes
1answer
129 views
Are $CAT(0)$-polygonal complexes median spaces?
A median space is a metric space $X$ for which for any three points $x, y , z \in X $ there exists a unique point $m$ such that $d(x,m)+ d(m, y)= d(x , y ), d(x,m)+ d(m, z)= d(x , z ), d(y,m)+ d(m, z)=...
1
vote
1answer
128 views
Approximate the following series on the euclidean grid
I had trouble in finding a closed form solution for the following series, so now I am trying to find a good approximation for it. The $\sqrt{i^2 + j^2}$ in the exponent comes from distances on the ...
2
votes
1answer
190 views
Why the VC dimension of triangles in 2D space is not greater than 7?
I understand that there are sets of 7 points on a circle that can be fully
shattered using triangles.But, it is not clear to me why it cannot shatter 8 points.
Is there any intuitive way of arriving ...
13
votes
1answer
273 views
Can the graph of a symmetric polytope have more symmetries than the polytope itself?
I consider convex polytopes $P\subseteq\Bbb R^d$ (convex hull of finitely many points) which are arc-transitive, i.e. where the automorphism group acts transitively on the 1-flags (incident vertex-...
9
votes
2answers
1k views
Do two new special points in any triangle exist?
There are some special points in any triangle, as Fermat point, symmedian point, incenter, Morley center, et cetera.
Let $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $...
2
votes
0answers
113 views
Lebesgue density theorem for “doubling uniformly covering collections of subsets”
I am looking for a version of Lebesgue density theorem that works when restricting to "good" collections of balls with respect to (not necessarily doubling) metric measure spaces. Specifically
Let $(...
8
votes
1answer
256 views
Intersection of nested open ball in complete metric spaces is nonempty?
My question is that whether the following statement is true or not.
In a complete metric space $(X, d)$, if a sequence of open balls $\{B(x_i, r_i)\}_{i=1}^\infty$ satisfies
$$
\exists \epsilon > ...
3
votes
1answer
61 views
Maximal edge length of symmetric polytopes
For me, a polytope is the convex hull of finitely many points. It is said to be vertex-transitive / edge-transitive if its symmetry group acts transitively on its vertices / edges. Let's call a ...
2
votes
0answers
116 views
A problem on real analysis related to Hilbert's fourth problem
This is an extensive re-write of a question I deleted and which had basically the same title.
Identify the cylinder $S^1 \times \mathbb{R}$ with the space of (co)oriented lines in the plane by ...
2
votes
1answer
251 views
About the growth rate of a group
Let $G$ be a f.g. group and $d$ be a word metric w.r.t. a symmetric generating set. For $g\in G$, define $|g|:=d(g,e)$, where $e$ is the group identity. For $k\in\mathbb N$, put
$$n_k:=\#\{g\in G: |g|...
19
votes
1answer
537 views
Diameter of a quotient of the infinite dimensional sphere
Suppose a group $\Gamma$ acts by isometries on the Hilbert space $\mathbb{H}^\infty$ and it fixes the origin. So $\Gamma$ acts on the unit sphere $\mathbb{S}^\infty$ as well.
Assume that the action $...
5
votes
1answer
83 views
Clustering distance
Is there a good notion of distance between partitions of a (fixed, finite) set? The context is this: suppose I have a clustering algorithm, which clusters points using some method or other. Now, I ...
12
votes
2answers
315 views
Geodesic current supported on a pencil?
Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the ...
11
votes
0answers
151 views
Elkies points in the plane of a triangle $ABC$
Noam Elkies proved that if $x,y,z$ are positive numbers, then there is a unique point $P$ inside $ABC$ such that the inradii $r_a,r_b,r_c$ of the triangles $BPC, CPA, APB,$ respectively, satisfy
$$ ...
14
votes
2answers
395 views
The Disco Ball Problem
Let me first give some of a background as to where I got this problem. I had a math teacher ask me a few months ago: "How many 1 unit by 1 unit squares could one fit on a sphere with a radius of 32 ...
6
votes
0answers
69 views
Hölder isoperimetric problem
Denote by $S_r$ the usual circle of radius $r$, with the path metric ($d(x,y) = r\theta$, where $\theta$ is the angle between the vectors $x$ and $y$), and let $\alpha \in (1/2,1)$. Consider the ...
1
vote
0answers
87 views
Checking Planar Convexity of 4 Points with Stewart's Formula
Is the following conjecture correct?
Conjecture:
If $A,B,C,D$ are four points in general position in the euclidean plane, with
$a:=\|C-B\|,\ \ b:=\|C-A\|,\ \ c:=\|B-A\|$
$a':=\|D-A\|,\...
2
votes
0answers
56 views
8-partition of a planar convex body by 4 concurrent lines
It is known1 that any convex body $K$ in the plane can be
partitioned into $6$ equal-area pieces by $3$ concurrent lines
which meet at a point in $K$.
Call this a $6$-partition.
This result cannot be ...
1
vote
1answer
159 views
A possible characterization of the cube?
Let $P$ be the $1$-skeleton of a convex polyhedron fixed in $\mathbb{R}^3$,
and $|P|$ the sum of the Euclidean lengths of the edges of $P$.
Let $P_1, P_2, P_3$ be the perpendicular projections of $P$
...
5
votes
1answer
236 views
A possible characterization of sphere or projective space
Is there a compact Riemanian manifold $M$ not diffeomorphic to sphere or real or complex or quaternion projective space which admit a diffeomorphism $f$ with the property that $$\forall x \in M, \...
