All Questions
Tagged with dg.differential-geometry mg.metric-geometry
617 questions
5
votes
0
answers
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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
0
answers
209
views
Stable norm on hyperbolic surfaces
For a hyperbolic surface $S$ and a homology class $h\in H_1(S)$ its stable norm is defined as $\lim_{n\to\infty}\frac{1}{n}l(nh)$, where $l(nh)$ means the minimal length among all closed geodesics ...
- 10.4k
3
votes
1
answer
109
views
Area of a sphere in Alexandrov spaces
Let $(X,d)$ be an $n (\geq2)$ dim Alexandrov space with curvature $\geq k$. $B(x,r)$ is an open ball in $X$. Let $M_{k,n}$ be the $n$ dim space form of constant curvature $k$. $B_k(r)$ is an open ball ...
- 169
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
3
votes
0
answers
261
views
Exponential map for non-smooth Finsler manifolds
Context
I'm interested in studying reversible Finsler manifolds which do not have the strong convexity of the Hessian property (that is the Finsler function is a regular norm on every tangent space). ...
- 5,405
5
votes
1
answer
1k
views
Smoothness of the square of the distance function on a Riemannian manifold
Let $(M^n,g)$ be a smooth Riemannian manifold. The distance between two points is the infimum of the lengths of the curves which join the points. Consider the square of the distance function
$d^2\...
- 59
5
votes
1
answer
425
views
Fundamental group of compact manifolds with non-negative Ricci curvature and Bieberbach theorem
Let $M$ be a compact n-dimensional Riemannian manifold with non-negative Ricci curvature. Then its universal cover $\tilde{M}$ is isometric to $\mathbb{R}^p\times N$ for some $p\leqslant n$ and $N$ ...
2
votes
1
answer
146
views
Prove a consequence of Poincare inequality and volume doubling
The question is Lemma 5.3 in [1] (with-out detailed proof). But I don't know how to prove.
Let $M$ be a (finite dim) manifold satisfying the following two assumptions:
(1) for any $x\in M$, and any ...
- 169
6
votes
0
answers
191
views
Cut locus on a hypercube
Inspired by the question, "Shortest path connecting two opposite points on a cube":
Q. What does the cut locus with respect to one corner of a hypercube
in $\mathbb{R}^d$ look like?
"The cut ...
- 151k
16
votes
1
answer
667
views
Can a shape rolling inside itself reproduce that shape?
Q. Is the circle the only shape that, when rolling inside itself,
has a point that draws out a scaled copy of itself?
Let $C$ be a simple, closed, smooth curve in the plane.
(Likely "smooth" can be ...
- 151k
7
votes
1
answer
259
views
Can we realize the smooth metric of an Alexandrov space with nonnegative curvature by a Riemannian structure?
We know that a smooth Riemannian manifold with nonnegative curvature is an Alexandrov space (with induced metric) of nonnegative curvature.
What about the converse? That is, given a smooth metric d ...
- 1,799
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
0
votes
1
answer
181
views
Convex planar curves and intersections [closed]
Given two planar regular convex not-closed curves C and C_1.
Let A the set of finite intersections between C and C-1.
Then what is the stricter upper bound of |A|?
I would say 2.
Thanks.
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
1
vote
1
answer
273
views
Regularity of a generalized polar coordinate metric with two angles
Flat space in polar coordinates takes the form
$$ds^2=dr^2+r^2d\phi^2$$
To avoid a conical singularity at the origin, we must impose that $\phi$ is periodic with period $2\pi$.
Now consider the ...
- 293
5
votes
0
answers
464
views
Examples of spiraling geodesics?
Does there exist a closed, bounded surface $S$ embedded in $\mathbb{R}^3$
that has a geodesic $\gamma$ that spirals around a point $x$, getting closer
and closer, but never reaching $x$?
Here I ...
- 151k
0
votes
0
answers
259
views
The analyticity of distance function
Given a real analytic compact manifold $M$ with boundary $\partial M$, suppose that $M$ is embedded in an open analytic manifold $N$ which has the same dimension as $M$. Is the distance function $d(x)...
- 103
4
votes
1
answer
161
views
Is a minimal surface $S$ that is bounded by an analytic closed curve $C$, analytic?
Let $C$ be an analytic closed curve (in the form of an unknot) in $\mathbb{R}^3$ and let $S$ be a minimal surface (a disk) bound by $C$. Is $S$ always analytic? Can you point out some references?
- 415
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
3
votes
1
answer
284
views
Cover sphere's surface with triangles of geodesic distance edges
Problem:
Given:
A set of points (their coordinates) on a sphere's surface.
Goal:
Connect them forming triangles, such that:
a)The union of triangles cover the whole surface of the sphere.
b)The ...
- 137
5
votes
1
answer
412
views
Continuous deformation of soap films
Let $S$ be a soap film bounded by an unknotted wireframe cycle (in $R^3$). Why is it the case that as we deform the wireframe in $R^3$, $S$ deforms continuously?
- 51
4
votes
3
answers
927
views
Lower bound for the normal injectivity radius
Let $(M,g)$ be a closed Riemannian manifold and let $N$ be a closed embedded submanifold. A tube $T(N,r)$ of radius $r$ of $N$ is defined as the set of points of $M$ which can be reached by a ...
- 131
3
votes
0
answers
262
views
Least reasonable regularity on Riemannian metric tensor to define a metric [closed]
We assume that $M$ is a smooth manifold of dimension $m$, and we assume that $g$ is a smooth Riemannian metric $g$ on $M$.
If $M$ is moreover path-connected, then $g$ induces a metric (in the sense ...
- 5,327
1
vote
0
answers
113
views
upper bound for heat kernel of Grushin operator
Let $\Omega$ be a bounded open domain in $\mathbb{R}^{n+1}$ with smooth boundary. $\Omega\cap\{(0,\cdots,0,y)\in\mathbb{R}^{n+1}| y\in \mathbb{R}\}\neq \varnothing$
Consider the sub-elliptic operator
...
- 287
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
3
votes
0
answers
127
views
Behaviour of geodesics on surfaces as one of the two endpoints moves slightly
Let $u$ and $v$ be two points on a surface (I guess, a Riemann surface) $\Sigma$ such that there is a unique geodesic between $u$ and $v$ on $\Sigma$. Now let $l$ be an arbitrary line that passes ...
- 415
6
votes
1
answer
148
views
Does a minimum area disk that is bounded by a cycle $C$ continuously deform in $R^3$ as $C$ moves in $R^3$?
Let $C_1=(v_1,v_2,\ldots,v_{i-1},v_i)$ and $C_2=(v_1,v_2,\ldots,v_{i-1},v'_i)$ be two cycles that are drawn in $R^3$ in the shape of an unknot (not knotted) with straight line segments as their edges (...
- 415
3
votes
1
answer
231
views
mean curvature on Finsler Manifolds
Let $F$ be a Finsler metric and $g$ a Riemannian metric for $M$. Is there on Finsler manifolds a similar curvature to the mean curvature of Riemannian manifolds, such that if $F=\sqrt{g}$ then both ...
- 61
2
votes
0
answers
191
views
Geometric properties of solutions of Hamiltonian system
Context : We are interested in the following dynamic with state $(q,\varphi)$
$$
\dot q = \varepsilon F(q,\varphi), \quad \dot \varphi = \omega(q) + \varepsilon G(q,\varphi)
$$
($\varepsilon >0$ ...
- 141
2
votes
0
answers
100
views
Is the cut locus of a compact subset in an Alexandrov space negligible?
Let $(M,d)$ be a compact, n dimensional Alexandrov space with curvature $\geqslant -1$ and without boundary. $\mathcal{H}^n$ is the n dimensional Hausdorff measure.
Let $K\subset M$ be a compact ...
- 21
5
votes
2
answers
546
views
Limit space of a sequence of Riemannian manifolds with uniformly bounded below Ricci curvature
Let $\{M^n_i\}_{i=1}^\infty$ be a sequence of closed smooth Riemannian $n$-dimensional manifolds with uniformly bounded below Ricci curvature and uniformly bounded above diameter. The Gromov ...
- 21.8k
8
votes
2
answers
378
views
Curves embedding in plane
Given two closed simple(no self-intersection point) curves $C_1,C_2$ in the plane $\mathbb R^2$, is there a good way to judge whether one curve can be embedded inside the other one, here embedding ...
- 1,915
5
votes
1
answer
1k
views
Does every smooth manifold admit a metric with bounded geometry and uniform growth?
Let $M$ be a smooth manifold, $g_M$ a Riemannian metric, and consider for $x\in M$ the volume growth function, $gr_x$ that maps $r>0$ to the volume $vol_{g_M}(B(x,r))$. My interest is to see ...
- 541
2
votes
0
answers
225
views
Negative-curvature behaviour of higher-rank symmetric spaces
Let $X$ be a symmetric space of noncompact type with $rk\ X\geq 2$ and let $G$ be the identity component of the isometry group. Pick points $p\in X$ and $\xi\in\partial_{\infty}X$, with $\xi$ regular, ...
5
votes
1
answer
238
views
The radially symmetric isoperimetric problem
A solution to the $l$-isoperimetric problem on a Riemannian surface $(M,g)$ is a smooth closed curve $\gamma \subset M$ of length $l$ which minimizes the isoperimetric constant:
$$h(\gamma) = \frac{l}{...
6
votes
0
answers
113
views
Does the Hodge *-operator act on the tangent space at 0 to the space of integral (n-1)-cycles in a conformal manifold of dimension d=2n?
Suppose $M$ is a compact, oriented conformal manifold of even dimension $d=2n$.
Write ${\cal D}^{\mathit{int}}_{k}(M)$ for the space of integral
$k$-currents in $M$
and write ${\cal D}^{\mathit{int}}...
- 351
4
votes
1
answer
239
views
Sufficient conditions for a curve on the sphere to be the Gauß map of a closed path
I was wondering which curves on the $n-1$ sphere arise as the Gauss maps of closed paths in $\Bbb R^n$. Necessary conditions are obviously that the path on the sphere is the image of some smooth $S^1\...
- 3,017
5
votes
2
answers
359
views
References for metrics in matrix groups
I am studying a very concrete matrix group with a riemaniann (right invariant) metric for solving a question on Applied Math. I need explicit formulas for the distance between two matrices, geodesics ...
2
votes
1
answer
135
views
Volume of the subelliptic ball
Let $\Omega \in \mathbb{R}^n$ a bounded open set when $n\geq 2$, and let $X_{1},X_{2},\cdots,X_{m}$ be real smooth vector fields that satisfy Hormander condition on $\Omega$. If we denote $Q(x)$ as ...
- 21
11
votes
4
answers
369
views
Tameness in $\mathbb{R}^{n^2}$ of the subset consisting of matrices of positive determinant
The Lie group $GL(n)$ being a manifold is locally path-connected. Consider its connected component of the identity $C\subseteq\mathbb{R}^{n^2}$. What is a good way of showing that $C$ is a tame ...
- 16.6k
6
votes
1
answer
1k
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Compact manifolds locally bi-Lipschitz to Euclidean space
I have a compact manifold $M$, and I am allowed to choose some Riemannian metric on it, exactly which I don't care. But I would love it if I could choose the metric $g$ such that every point has an ...
- 35.5k
7
votes
2
answers
436
views
Are square tiled surfaces dense in the moduli space of translation surfaces?
I'm reading the survey "An introduction to Veech surfaces" by Pascal Hubert and Thomas Schmidt.
At page 19 they state "In any fixed stratum, the set of square-tiled surfaces of that stratum is dense....
- 73
4
votes
1
answer
197
views
Distance comparison in submanifold versus in the underlying manifold
Let $(M,g)$ be the (underlying) manifold, $(S,g|)$ be a submanifold. Let $a,b,c \in S$. It's not in general true that $d_M(a,b)\leq d_M(a,c) \implies d_S(a,b)\leq d_S(a,c)$.
QUESTION I:
The above ...
- 1,467
22
votes
1
answer
1k
views
Just how close can two manifolds be in the Gromov-Hausdorff distance?
Suppose that we have two compact Riemannian manifolds $(M,g)$ and $(N,h)$. Define the Gromov-Hausdorff distance between them in your favorite way, I'll use the infimum of all $\epsilon$ such that ...
- 646
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
8
votes
1
answer
600
views
Questions on Thurston's metric on Teichmüller space
I'm reading the famous "Minimal stretch maps between hyperbolic surfaces" by William Thurston and I'm trying to understand the key theorem 8.1. I have many unclear points so I hope someone can help me ...
- 81
4
votes
1
answer
870
views
Applying Cheeger and Colding segment inequality
The question turns out quite long and maybe a bit vague, I apologize in advance for that.
I am currently trying to understand Cheeger and Colding proof of the almost splitting theorem. Currently I am ...
- 4,123
9
votes
1
answer
321
views
Convex body with affine-equivalent cross-sections
I recently discovered the following fact: Let $K\subset\mathbb R^3$ be an origin-symmetric convex body with smooth and strictly convex boundary. Suppose that all central cross-sections of $K$ (that is,...
- 32.4k