All Questions
63 questions
0
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0
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33
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Request for resources on directional derivative of the Riemannian distance function, and Berger's lemma about geodesics realizing the diameter
I've been recently interested in directional derivatives of the Riemannian distance function, and I came across this question, and its answer by Sergei Ivanov, where he stated an important result: (I ...
- 1,467
5
votes
0
answers
78
views
Is there a generalization of the Diameter Sphere Theorem to orbifolds?
The Diameter Sphere Theorem of Grove and Shiohama asserts that if $M$ is a compact Riemannian manifold with sectional curvature bounded from bellow by 1 and diameter greater than $\pi/2$, then $M$ is ...
- 51
6
votes
1
answer
207
views
Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces
I am new into geometric group theory and I have recently started reading the book "Sur les Groupes Hyperboliques d’après Mikhael Gromov" by Ghys and de la Harpe. The following inequality ...
- 101
5
votes
1
answer
155
views
Variants of the Bonk-Schramm embedding
Recently I heard about the following embedding theorem of Bonk and Schramm: every Gromov hyperbolic geodesic metric space with "bounded growth" is roughly similar to a convex subset of $\...
5
votes
1
answer
530
views
Geodesic distance on $\mathrm{SO}(n)$
$\DeclareMathOperator\SO{SO}$Recently I came across this old MSE post or this paper (w.o. proof) discussing the geodesic distance on $\SO(n)$ when it is equipped with the left-invariant Riemannian ...
- 364
11
votes
3
answers
1k
views
What is the minimum-curvature curve interpolating a given set of points in the plane?
We are given a set $X$ of $n\ge 3$ points in $\mathbb{R}^2$, belonging to the boundary of the convex hull of $X$ itself. Let $\Gamma(X)$ be the set of all convex, simple closed curves in $\mathbb{R}^2$...
- 1,677
2
votes
0
answers
65
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Connection between a function and its usage in geometry [closed]
I know nothing about geometry, but I found a function which seems to have something to do with geometry.
This function is, $$f(x,y,z) = \dfrac{(x,y,z)}{\sqrt{1 + x^2 + y^2 + z^2}}$$
where $x,y,z$ is ...
- 121
4
votes
0
answers
186
views
Ends of a negatively curved Riemannian manifold
Let $M$ be a complete Riemannian manifold. Let us use the standard definition of "end", for example, as in this article. If $M$ has non-negative Ricci curvature, it is well-known that it has ...
- 41
0
votes
0
answers
425
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Compact connected Riemannian manifolds are Ahlfors regular metric space
Let $(M,g)$ be a compact connected $n$-dimensional Riemannian manifold; let $(X,d)$ denote its associated metric (length) space. A comment on the original formulation of this post mentioned that $(X,...
- 5,405
1
vote
0
answers
101
views
Actions of finite groups on compact symmetric spaces
I am interested in series of finite subgroups of the classical compact simple Lie groups which have big orbits on compact symmetric spaces and where the double coset space has some nice explicit ...
- 8,597
7
votes
2
answers
359
views
Cone unfolding of space curves
There is a natural length-preserving operation which transforms any rectifiable space curve $\gamma\colon [a,b]\to R^n$ into a planar curve $\tilde\gamma \colon [a,b]\to R^2$. This operation, which ...
- 7,169
6
votes
2
answers
317
views
Quasi-isometric embedding of graphs in non-compact riemannian surfaces
Given a complete riemannian surface $(S,m)$, where $S$ is homeomorphic to $\mathbb{R}^2$, I would like to find a weighted graph $G$ (which means a graph with real non-negative weights on the edges), ...
- 935
5
votes
0
answers
272
views
When do surfaces in $\mathbb{R}^4$ intersect all their translations in one direction?
I am looking for research or references on the following problem.
Let $S$ be a smoothly embedded connected surface in $\mathbb{R}^4$, with or without boundary. Fix some axis in $\mathbb{R}^4$, let $d ...
- 1,763
7
votes
1
answer
483
views
Furthest distance half the diameter?
Let $S$ be the surface of a convex body, polyhedral or smooth,
embedded in $\mathbb{R}^3$.
For a point $x \in S$, let $F(x)$ be the set of furthest points
from $x$, measured by shortest paths on the ...
- 151k
3
votes
0
answers
159
views
Upper bound on the geodesic distance in a Lipschitz domain
I was wondering if the following result is true. If yes, could you please suggest a reference. The result seems to have been used at several papers without quoting any reference. Is the proof ...
- 895
14
votes
4
answers
963
views
Steiner's inequality reference request
I remember seeing somewhere that for every connected compact set $\Omega$ in $\mathbb{R}^2$ with piecewise $C^1$ boundary we have
$$A(\Omega_r)\leq A(\Omega)+L(\partial \Omega)r+ \pi r^2,$$
where
$$\...
- 295
1
vote
0
answers
162
views
Gromov-Hausdorff relative compactness without curvature restrictions
A famous theorem of Gromov says that the set of compact Riemannian manifolds with $Ric \geq c$ and $\text{diam} \leq D$ is relatively compact in the Gromov-Hausdorff metric. Chapter 10 of the book by ...
- 1,407
17
votes
5
answers
883
views
Rigidity of convex polyhedrons in $\mathbb R^3$ with faces removed
Take a convex polyhedron $P$ in $\mathbb R^3$ and remove all the faces, i.e. leave only the edges. Call this graph $E$. Let us now try to continuously deform $E$ in $\mathbb R^3$ so that all the edges ...
- 14.3k
3
votes
1
answer
704
views
Bishop-Gromov volume comparison on manifolds with negligible negative Ricci curvature
Let us consider a complete Riemannian manifold $M$ of dimension $n$ with $Ric \geq 0$. Then the Bishop-Gromov volume comparison theorem says that for any $p \in M$, the function
$$ \frac{\text{Vol}(B(...
- 31
8
votes
1
answer
400
views
Multidimensional gluing theorem for Riemannian manifolds
I would like to understand whether the following multidimensional (partial) generalization of the A.D. Alexandrov gluing theorem is true and, if yes, whether there is a reference.
(The original ...
- 21.8k
22
votes
2
answers
1k
views
Why doesn't this construction of the tangent space work for non-Riemannian metric manifolds?
In the 1957 paper, On the differentiability of isometries, Richard S. Palais gives a way to construct the tangent spaces of a Riemannian manifold using only its metric space structure (Theorem, p.1).
...
- 2,680
5
votes
0
answers
1k
views
Reach of manifold vs. $C^k$-manifold
The reach $\tau_M$ of a manifold $M$ is the largest number such that any point at distance less than $\tau_M$ from $M$ has a unique nearest point on $M$.
This concept seems quite related to the local ...
- 151k
6
votes
2
answers
381
views
Sources for Alexandrov surfaces
There are two distinct notions in differential geometry associated
with A. D. Alexandrov: (1) Alexandrov spaces of courvature bounded
from below; (2) Alexandrov surfaces of bounded total curvature (...
- 16.6k
0
votes
0
answers
61
views
Heat trace asymptotic coefficients for conformal metrics $\widetilde{g}=e^{f}g$ surfaces
As is well known $\sum e^{-\lambda_{k}t}\approx(4\pi t)^{dim(M)/2}\sum a_{j}t^{j}$, where $a_{j}$ are geometric properties of manifold M.
Moreover, the arbitrary order coefficients don't have closed ...
- 5,474
14
votes
3
answers
963
views
Conjugate points on cut locus
Let $M$ be a Riemannian with nonempty boundary $\partial M$.
Define multiplicity of $x\in M$ as the number of minimizing geodesics from $x$ to $\partial M$.
The following fact seems to be standard:
...
- 45k
2
votes
0
answers
127
views
Functional inequality under mean curvature flow
Let $\Sigma$ be a hypersurface in $\mathbb R^n$ and $\Sigma_t$ be a variation of $\Sigma$ under the mean curvature flow under an extra condition that ${\rm vol}_{n-1}(\Sigma)={\rm vol}_{n-1}(\Sigma_t)$...
- 143
8
votes
1
answer
682
views
Geometry of convex sets in Riemannian manifolds
Let $M$ be a smooth Riemannian manifold without boundary. Let $X\subset M$ be a closed subset which is a smooth submanifold with boundary, $\dim X=\dim M$. Assume that $X$ is locally convex, i.e. any ...
- 21.8k
5
votes
0
answers
391
views
Gage-Grayson-Hamilton curve-shortening flow, at an angle
The Gage-Grayson-Hamilton curve-shortening flows along the normal to the curve:
&...
- 151k
5
votes
1
answer
328
views
Is a space with p-norm a Finsler manifold?
Suppose $\mathbb{R}^n$ is equipped with the p-norm $\left\Vert x \right\Vert_p$. Let $x\in \mathbb{R}^n$ and let $y$ be in a neighborhood of $x$. The distance between $x$ and $y$ can be defined as $\...
- 51
1
vote
1
answer
145
views
Continuity of Busemann-Hausdorff area density
I am trying to find out why the Busemann-Hausdorff area density as defined by Burago and Ivanov is continuous. Here, $GC_m(V)\subset \Lambda^m(V)$ denotes the simple $m$-vectors in an $n$-dimensional ...
- 193
3
votes
1
answer
205
views
Reference: Finsler Derivative?
On the wikipedia page "Generalizations of derivative" the author mentions: " in Finsler geometry, one studies spaces which look locally like Banach spaces. Thus one might want a derivative with some ...
- 5,405
8
votes
1
answer
911
views
Avoiding mean-curvature flow dumbbell neck-pinch by inflating a surface
It is well known that
Grayson's dumbbell neck-pinch1,2 separates
into disconnected pieces under
mean curvature flow:
Image ...
- 151k
2
votes
1
answer
232
views
Shortest paths in Alexandrov spaces
Let $X$ be an Alexandrov space with curvature bounded from below (if necessary, $X$ might be assumed to be finite dimensional or even compact).
Question 1. Is it true that every point of $X$ has a ...
- 21.8k
5
votes
1
answer
906
views
Boundaries of relatively hyperbolic groups
When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups $\...
- 10.4k
9
votes
1
answer
3k
views
Oloid and sphericon: rolling develops entire surface
Wikipedia says that,
"The oloid is one of the only known objects, along with some members of the sphericon family, that while rolling, develops its entire surface."
Below are illustrations of ...
- 151k
15
votes
0
answers
382
views
Has Cheeger's 'de Rham cohomology' of metric measure spaces been studied beyond its definition?
In J. Cheeger's 'Differentiability of Lipschitz Functions on Metric Measure Spaces' (Geometric and Functional Analysis, 1999, Vol. 9 pp 428-517, see here), a 'de Rham cohomology group' $H_{dR}^1(Z,\mu)...
user14166
3
votes
1
answer
292
views
Existence of Simple Closed Straightest Geodesics
There are at least three distinct simple closed quasigeodesics on convex polyhedra [Mat. Sb. (N.S.), 1949, 25(67) :2, 275–306 Quasi-geodesic lines on a convex surface Pogorelov].
Is the same true ...
- 133
16
votes
2
answers
2k
views
There are two points on the Earth's surface that ... ?
At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...?
What is the strongest, most impressive statement one can make here? The ...
- 151k
14
votes
1
answer
1k
views
Egg-ovoid rolling down an inclined plane
I am seeking a mathematical analysis of an egg-ovoid rolling down an inclined plane,
for pedagogical reasons.
It is well-known folk lore that the shape of an egg prevents it from rolling away from
...
- 151k
15
votes
3
answers
4k
views
About MF Atiyah and R Bott's 1983 paper
I am a theoretical physics major student working on string theory. I want to understand the work of MF Atiyah and R Bott, "The Yang-Mills equations over riemann surfaces" . What kinds of mathematical ...
- 395
24
votes
5
answers
4k
views
Weitzenböck Identities
I asked this question at Maths Stack Exchange, but I haven't received any replies yet (I'm not sure how long I should wait before it is acceptable to ask here, assuming there is such a period of time)....
- 19.3k
19
votes
1
answer
2k
views
Does this Banach manifold admit a Riemannian metric?
First, the question; after, the motivation.
Consider 27.6 (pdf pp. 262-263) in The convenient setting of global analysis (AMS, 1997), and, in particular, the example given at the end of it, which ...
- 7,831
17
votes
1
answer
526
views
Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space?
Let $X$ be a complete CAT$(0)$ metric space, and $\partial X$ its boundary.
One way to define $\partial X$ is as the equivalence class of geodesic rays
$\gamma(t), \gamma'(t)$
that remain within a ...
- 151k
24
votes
1
answer
1k
views
Non-regular Connected Hausdorff Banach Manifold
After reading this MO post, I am wondering:
Is every (connected) Hausdorff Banach manifold a regular space?
Though unjustified, page 53 of this paper nonchalantly states: "Note that a Hausdorff ...
- 7,831
10
votes
1
answer
2k
views
Curves of constant curvature on an ellipsoid
It is not difficult to see that the curves of constant geodesic curvature on a geometric sphere
are all circles: simple, closed curves that are geometric circles lying in a plane:
&...
- 151k
1
vote
0
answers
371
views
Simple development of simple curve on a cone
Let $\Lambda$ be a cone with apex $a$ and apex angle $\alpha$. Draw a simple (non-self-intersecting)
curve $C=(x,y)$ on $\Lambda$, and then develop it to a curve
$\overline{C}$ on a plane by rolling $...
- 151k
5
votes
0
answers
1k
views
"The famous Lusternik-Schnirelmann Theorem of the Three Closed Geodesics"
The title is a quote from p.256 of Wilhelm Klingenberg's 1995
Riemannian Geometry (Google Books link):
Every surface homeomorphic to a sphere $\mathbb{S}^2$ has three distinct, simple, closed ...
- 151k
2
votes
1
answer
308
views
Connecting tangents of convex curves: at some point orthogonal?
Let $a(t)$ and $b(t)$ be two smooth, nested convex curves in the plane, $t\in[0,1]$:
Suppose the parametrization of $a()$ and $b()$ is such that $\dot{a}(t)$ is ...
- 151k
5
votes
0
answers
350
views
Areas dominated by two points on a surface: Equal?
Let $S$ be a smooth compact surface in $\mathbb{R}^3$, with two distinct, distinguished points
$a,b \in S$. Let $R(a)$ be all the points of $S$ closer to $a$ than to $b$, and $R(b)$ all the
points of ...
- 151k
28
votes
2
answers
3k
views
Probing a manifold with geodesics
Supposed you stand at a point $p \in M$ on a smooth 2-manifold $M$
embedded in $\mathbb{R}^3$.
You do not know anything about $M$.
You shoot off a geodesic $\gamma$ in some direction $u$,
and learn ...
- 151k