All Questions
Tagged with dg.differential-geometry mg.metric-geometry
617 questions
6
votes
3
answers
365
views
Sliding through a curvature-bounded tube: Maximum volume?
My 1st question has a straightforward answer but I'd appreciate hints on a proof. My 2nd question is open from my point of view.
Q1. Is it the case that the maximum convex volume body inside a ...
- 151k
13
votes
1
answer
550
views
Regularity of geodesics
If $M$ is a graph of a $C^1$ function $f:\mathbb{R}^n\to\mathbb{R}$, is it true that the length minimizing geodesics on $M$ are $C^1$? I expect a counterexample.
For a related discussion see Metric ...
- 28k
4
votes
0
answers
756
views
Tangent space and gradient on subspace of Wasserstein space given by finitely supported measures
Let $\mathcal{P}_2(M)$ be the 2-Wasserstein space over some Riemannian manifold $(M,g)$ (connected, complete, and without boundary). Let $\mathcal{FP}_2(M,n)$ be the subspace of probability measures ...
- 1,675
3
votes
0
answers
106
views
Bound on change in relative length from 'well-behaved' Jacobian?
(This question was originally asked on Mathematics Stack Exchange, and sat there for several weeks with low views and no answers.)
Let $\phi$ and $\gamma$ be rectifiable curves in the same length ...
- 290
3
votes
1
answer
184
views
Relation between a distance function and normal coordinations
$
\newcommand{dist}{\operatorname{dist}}
\newcommand{B}{\mathbb{B}}
$
Let $\mathcal {M}$ be a Riemannian manifold, $p \in S \subset \mathcal{M}$ and $r>0$. Denote $S_{r} := S \cap\B(p,r)$.
...
- 1,991
8
votes
1
answer
412
views
Homeomorphism/ homotopy types of non-negatively curved manifolds
A (special case of a) theorem of Gromov says for any $n\in \mathbb{N}$ there exists a constant $C(n)$ such that for any smooth connected closed $n$-dimensional Riemannian manifold with non-negative ...
- 21.8k
6
votes
1
answer
254
views
Triangulations of convex surfaces
Let $M$ be a smooth closed positively curved surface in Euclidean 3-space, $T$ be a geodesic triangulation of $M$, and $E$ be the edge graph of the convex hull of vertices of $T$.
It is easy to see ...
- 7,149
8
votes
4
answers
516
views
Must a bending of the cylinder leave the bases planar?
Set $M=\{(\cos(\theta),\sin(\theta),z):\theta\in[0,2\pi],z\in[0,1]\}$. A bending of $M$ is a smooth map $\Gamma:M\times [0,1]\rightarrow \mathbb{R}^3$ such that
1) $\Gamma[M\times\{t\}]$ is a ...
- 1,117
3
votes
0
answers
108
views
Radial Poincare inequality for Gaussian measures
Let $\mu$ be a zero mean Gaussian probability measure on $\mathbb{R}^n$ whose covariance is less than the identity. If $f$ is a $1$-Lipschitz real function on $\mathbb{R}^n$ such that there exists a ...
- 2,772
1
vote
1
answer
185
views
A possible characterization of Euclidean geometry via the curvature of the Median-submanifold
Is there a Riemannian metric $g$ on $\mathbb{R}^2$ with inducing distance
$d$ which is not isometric to the standard metric but satisfy the property quoted bellow?
For every two distinct ...
- 356
4
votes
0
answers
100
views
Is there a fiber bundle for Alexandrov spaces collapsing to a manifold?
Let $\Psi(i)\to 0$ as $i\to \infty$.
Let $A_i$ be a sequence of n dimensional Alexandrov spaces with curvature $\geqslant k$, that Gromov Hausdorff converge to an m dimensional closed Riemannian ...
1
vote
0
answers
43
views
Quantitative error control in Minkowski-Stein formula
Let $K\subseteq\mathbb R^d$ be a compact convex body with non-empty interior, and $E$ be a $(d-1)$-dimensional linear subspace of $\mathbb R^d$. Let $\theta\in\mathbb R^d$ be the unit vector such that ...
- 373
4
votes
2
answers
266
views
Metrics with fixed conformal structure and diameter
I have three questions.
I consider a sequence of metrics $h_n$ on a two-dimensional torus which all induce the same conformal structure. Suppose that the volume of $h_n$ is always $1$. Is it possible ...
- 2,696
7
votes
1
answer
637
views
What is the distance between two points on the Berger metric of the squashed three-sphere?
The Berger metric on a "squashed" three-sphere is given (in Euler coordinates) by
4 $ds^2 = \lambda^2 (d \tau + \cos \theta d \phi)^2 + d \theta^2 + \sin^2 \theta d \phi^2$.
See for example Eq. 1....
- 273
2
votes
0
answers
108
views
Does a smooth dynamical system always come with a metric
Warning: My education in formal mathematics is very weak so I apologize for any confusions/errors in the following, please don't hesitate to correct me.
Question: Consider a smooth dynamical system $...
- 21
6
votes
0
answers
369
views
Distance measures that preserve Pythagoras' theorem but break the triangle inequality
In information geometry, we can think of the Kullback-Leibler divergence as being "something like a squared distance."
The sense of this is that if we have three probability measures, $P$, $Q$ and $R$...
- 1,344
5
votes
0
answers
141
views
Yang-Mills connection on circle bundle
Let $(M,g)$ be a connected, oriented, compact Riemann surface with positive constant curvature $K_g$, and let $P\to M$ be a principal $S^1$-bundle on $M$. Can we find a Yang-Mills connection $A$ on $P$...
- 83
15
votes
1
answer
556
views
Characterization of a sphere: every "sub-sphere" has two centers
Let me ask this question without too much formalization:
Suppose a smooth surface $M$ has the property that for all spheres $S(p,R)$ (i.e. the set of all points which lie a distance $R\geq 0$ from $p ...
- 1,461
6
votes
1
answer
281
views
Convex sets in Alexandrov spaces
Let $X$ be a compact Alexandrov space with $curv\geq 1$ (and without boundary). Does $X$ always have a nontrivial compact convex subset without boundary?
Definition of a convex subset: $A\subseteq X$ ...
- 377
0
votes
1
answer
126
views
What’s the form of Gram matrix for right-angled hexagon
Informally, right-angled hyperbolic hexagon is a hyperbolic triangle with vertices outside infinity. I think there should be a Gram matrix for it, and what does it looks like?
(The Gram matrix here ...
- 21
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
3
votes
1
answer
180
views
Are the Euler Characteristics of noncollapsed manifolds with bounded Ricci curvature uniformly bounded above?
For positive number $C>0$, $d>0$, are the Euler Characteristics of n dimensional closed Riemannian manifolds $M$ with diameter $\leqslant d$, $|Ric|\leqslant C$ uniformly bounded?
If this is ...
4
votes
1
answer
147
views
Explicit examples of warped products Gromov converge to a cone
It's well known that a sequence of two dimensional Riemannian manifolds with uniform sectional curvature lower bound can Gromov-Hausdorff converge to a cone.
Let $y=|x|$, by rotating around the y-...
3
votes
0
answers
1k
views
Justification for using polar coordinate with Riemannian metric
Let $(M,g)$ be a $d$ dimensional Riemannian manifold, $\exp_p:V_p\to M$ the exponential map from a neighborhood of $0$ in $T_pM$ into $M$. Recall that we may assume $\exp_p$ maps $V_p$ ...
- 1,245
2
votes
0
answers
784
views
Is the Eguchi-Hanson metric a metric on $\mathbb{R}\times S^3$ or tangent bundle of $S^2$?
In the paper "Asymptotically flat self-dual solutions to euclidean gravity" by T.Eguchi and A.J. Hanson, the author constructed a Ricci flat metric on $\mathbb{R}\times S^3$. Let
$$
\sigma_x=\frac12(-...
0
votes
1
answer
362
views
Is the intersection of two minimal surfaces minimal?
Consider two $n$-dimensional minimal surfaces without boundary. Suppose they are embedded in some $\mathbb{R}^m$ in such a way that they intersect transversally. Is their intersection a minimal ...
- 1,703
18
votes
1
answer
901
views
How to compute the Gromov-Hausdorff distance between spheres $S_n$ and $S_m$?
Can we compute the Gromov-Hausdorff distance $d(\mathbb{S}_n,\mathbb{S}_m)$ for two different spheres $\mathbb{S}_n$ and $\mathbb{S}_m$, $m\neq n$? We consider the spheres with the metrics induced by ...
- 697
17
votes
5
answers
883
views
Rigidity of convex polyhedrons in $\mathbb R^3$ with faces removed
Take a convex polyhedron $P$ in $\mathbb R^3$ and remove all the faces, i.e. leave only the edges. Call this graph $E$. Let us now try to continuously deform $E$ in $\mathbb R^3$ so that all the edges ...
- 14.3k
3
votes
1
answer
704
views
Bishop-Gromov volume comparison on manifolds with negligible negative Ricci curvature
Let us consider a complete Riemannian manifold $M$ of dimension $n$ with $Ric \geq 0$. Then the Bishop-Gromov volume comparison theorem says that for any $p \in M$, the function
$$ \frac{\text{Vol}(B(...
- 31
13
votes
2
answers
872
views
Intrinsic vs Extrinsic geometry of convex surfaces
By Alexandrov's isometric embedding theorem, any locally convex metric prescribed on the sphere admits a realization as a convex surface in Euclidean 3-space, which, by Pogorelov's rigidity result, is ...
- 7,149
80
votes
1
answer
3k
views
Converse to Euclid's fifth postulate
There is a fascinating open problem in Riemannian Geometry which I would like to advertise here because I do not think that it is as well-known as it deserves to be. Euclid's famous fifth postulate, ...
- 7,149
7
votes
2
answers
243
views
Length of simple closed curve in half-translation surface
Let $R$ be a Riemann surface of genus $g\ge 2$ and $q$ an holomorphic quadratic differential on $R$. Together they determine a semi-translation structure: an atlas on $X$ such that its changes of ...
- 609
4
votes
1
answer
124
views
Convex caps with prescribed edges and curvature
Let $G$ be the edge graph of a convex subdivision of a convex polygon $P$ in the plane. I would like to construct a convex polyhedral cap $C$ (with zero boundary values) over $P$ whose edges project ...
- 7,149
7
votes
1
answer
438
views
An isoperimetric type of inequality in terms of Wasserstein distance/Optimal transport
Let $A \subset \mathbb{R}^n$ be a region having the same volume as an $n$ dimensional ball $B^n_R$ with radius $R$ centring at the origin.
Isoperimetric inequality says:
$ Vol_{n-1} \partial A \geq ...
- 365
16
votes
2
answers
756
views
Can a sphere glued into a soft 3d-mattress rotate continuously? (manifolds, SU(2) and the belt trick)
The question is triggered by the wonderful animations by Jason Hise:
https://www.youtube.com/watch?v=LLw3BaliDUQ
https://www.youtube.com/watch?v=6Ul_-ABYaYU
https://www.youtube.com/watch?v=...
0
votes
0
answers
109
views
Bounding Riemannian Distance
If $(M,g)$ is a geodesically complete Riemannian manifold of negative sectional curvature bounded below by $K<0$ then is it true that for any $x,y,x_0\in M$
$$
d_H(
Log(x_0,x),Log(y,x_0)
)
\leq
d_M^...
- 5,405
7
votes
0
answers
119
views
Approximating manifolds with boundary by closed ones
Fix numbers $n\in \mathbb{N},d>0,k\in\mathbb{R}$. Do there exist numbers $N\in\mathbb{N},K\in\mathbb{R}$ depending on $n,d,k$ only with the following property:
For any compact smooth Riemannian $n$...
- 21.8k
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
7
votes
1
answer
162
views
Estimate of number of boundary components of a compact Riemannian 2-surface
Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $-1$ and the diameter is at most $D$. Assume that near the boundary the ...
- 21.8k
7
votes
1
answer
231
views
Estimate of area of 2-dimensional surface
Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $\kappa$, the diameter is at most $D$, and the second fundamental form ...
- 21.8k
8
votes
1
answer
265
views
Isoperimetric inequality on the plane
Let $A$ be a connected compact domain with smooth boundary in the Euclidean 2-plane. Assume its diameter is at most $d$. Assume that the second fundamental form of the boundary is at most $-c$ where $...
- 21.8k
7
votes
2
answers
460
views
Gaussian Surface Area of Positive Semidefinite Cone
Let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e.g., one that has smooth boundary or is convex. We define the $\epsilon$-neighbor of $A$ in the ...
- 1,127
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
2
votes
0
answers
106
views
The dimension of the subspace of flat spin connections
I am interested in the the flat spin connections in a Riemann spacetime of dimension 4. They appear in the context of the frame formalism of metric gravity theories. I believe that they form a ...
- 51
4
votes
0
answers
207
views
Integral of second fundamental form
Let us have Riemannian manifold $M$ with boundary $N.$ Let $F$ be an immersion, such that $F:N\to M$ and $B$ be a second fundamental form on $N$ relative to $F.$ And let $f$ be a function on $N.$
...
- 41
3
votes
1
answer
112
views
Balanced polygons
Can someone explain to me what balanced polygons are and how they are used in calculations?
Particularly in Gerver's Sofa, Here is the link for the research, page 16
Thanks in regard
user108974
5
votes
1
answer
331
views
Distance function on a curve on a manifold
Suppose that we are given a non-negative even function $b\in C^\infty[-1,1]$ satisfying $b(0)=0$, $\sqrt{b(x+y)}\le \sqrt{b(x)}+\sqrt{b(y)}$ for any $x,y\in[-\frac12,\frac12]$. Can we always find a 3-...
- 187
9
votes
2
answers
299
views
Isoperimetric dimension for any (metric) measure space?
$\newcommand{\v}{\operatorname{vol}}$The isoperimetric dimension is the maximum $d$ s.t.
$$\v(D)\leq C\cdot \v(\partial D)^{d/d-1}$$
for all open with smooth boundary $D\subset M$, differentiable ...
- 5,474
12
votes
3
answers
988
views
Primary definition of a geodesic
I am wondering if there is a sense in which one of these definitions
for a geodesic on a smooth Riemannian manifold is primary to the other.
A geodesic has acceleration zero, i.e., it is self-...
- 151k
2
votes
0
answers
152
views
Do Alexandrov spaces with non-empty boundary satisfy $RCD^*$ condition?
Let $M$ be an $n-$ dim compact Alexandrov space with curvature $\geq k$ with non-empty boundary $\partial M$.
Recently, a notion of generalized lower Ricci curvature bound on metric measure spaces ...
- 169