All Questions
Tagged with dg.differential-geometry mg.metric-geometry
617 questions
10
votes
2
answers
926
views
Is there a volume-preserving diffeomorphism of the disk with prescribed singular values?
This is a cross-post. While working on a variational problem, I have reached to the following question.
Let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1\sigma_2=1$, and let $D \subseteq \mathbb{R}^2$...
- 6,741
4
votes
1
answer
299
views
Bending the hemisphere
Let $S^+$ be the upper hemisphere of the standard sphere in $\mathbb R^3$ and $b$ -- the boundary of $S^+$ (the equator).
Let $b'$ be a small isometric deformation of $b$ (a nearby curve of the same ...
- 2,934
1
vote
0
answers
79
views
A Lipschitz limit of Riemannian metrics with curvature in $[-1,1]$
Let $(M,g)$ be a compact manifold with a metric $g$ (not necessarily Riemannian one). Suppose that $(M,g)$ is a Lipschitz limit of a sequence of Riemannian manifolds $(M_n,g_n)$ with the sectional ...
- 14.3k
4
votes
0
answers
68
views
Lipschitz distance on the moduli space of compact Riemann surfaces of curvature $-1$
Recall, that for two metric spaces $X$ and $Y$ the Lipschitz distance between $X$ and $Y$ is the infimum over all bi-Libschitz maps $f:X\to Y$ of
$$\log(\max({\rm dil}(f),{\rm dil}(f^{-1}))).$$
Here ...
- 14.3k
3
votes
0
answers
93
views
What can be reflected by the $C^0$-limit of Riemannian metrics?
Let $M^n$ be a closed connected smooth manifold and {$g_i$} be a family of smooth Riemannian metrics on it such that $g_i$ $C^0$-converges to the smooth Riemannian metric $g$ on $M^n$.
Can it ...
- 1,799
1
vote
1
answer
237
views
A basic question about compact $C^1$ surfaces with boundary
Let $S \subset \mathbb{R}^3$ be a compact and locally $C^1$ simply-connected surface with a $C^1$ boundary with no self intersection. Is there a $C^1$ bijection $F: \overline{B(0,1)} \rightarrow \...
- 434
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
0
votes
0
answers
88
views
Bound on the distance from points to the boundary of a hyperbolic surface
Fix $\epsilon\in\mathbb{R}_{>0}$, $\Sigma$ a surface with boundary and let $\mathcal{T}_{\Sigma}(L_{1},...,L_{n})$ denote the Teichmüller space of hyperbolic structures of $\Sigma$ with geodesic ...
- 81
9
votes
1
answer
255
views
On the diameter of left-invariant sub-Riemannian structures on a compact Lie group
Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$ of dimension $m$.
We fix an inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak g$.
We may assume (in case is necessary) ...
- 2,446
5
votes
3
answers
525
views
Weyl tube formula for manifolds with boundary
In 1939 H. Weyl proved the following non-trivial theorem. Let $(M^n, g)$ be a closed smooth Riemannian manifold. Fix an isometric imbedding $\iota\colon M\to \mathbb{R}^N$ into a Euclidean space (now ...
- 21.8k
3
votes
0
answers
531
views
Geodesics (Local vs Global)
Let $M$ be a Riemannian manifold, and $p,q\in M$ two points. Now, if $M$ is a complete metric space the Hopf–Rinow theorem ensures that there is a geodesic $C\subset M$ joining $p$ and $q$ that ...
user75933
5
votes
0
answers
137
views
"Inflating" a closed, defined metric, manifold
Consider a two-dimensional Riemannian manifold homeomorphic to the sphere, with a defined metric.
Since we do not suppose that manifold to have a positive curvature,
we are not in the hypotheses of ...
- 51
3
votes
1
answer
144
views
Global conformal Killing vector fields in a Riemannian manifold
Suppose $t$ is a globally smooth Killing coordinate function in $(M,g)$ such that $\partial_t$ is a Killing vector field. This gives rise to an embedding $\mathbb R \times \Omega$ for the manifold $M$....
- 4,135
2
votes
1
answer
93
views
In a manifold, $\angle xpy>\frac{\pi}{2}$, for $q$ on $px$ or $py$, $B_q(r)$ homeomorphic to $B_p(r)$?
Let M be an n-dimensional Riemannian manifold without boundary, with sectional curvature $\geqslant -1$. For a point $p\in M$, suppose there exist $l, \delta>0$, $x,y \in M$ with $d(p,x),d(p,y)>...
6
votes
1
answer
310
views
Asymptotic bound on minimum epsilon cover of arbitrary manifolds
Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\varepsilon)$ denote the minimal cardinal of an $\varepsilon$-cover $P$ of $M$; ...
- 71
2
votes
1
answer
215
views
Does fractallity depend on the Riemannian metric?
Edit: According to comment of Andre Henriques we revise the question:
In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that ...
- 356
4
votes
1
answer
369
views
Comparing two Riemannian metrics on Grassmannian
Let $G_r(n)$ be the real Grassmannian which is the collection of all $r$ dimensional subspace in $\mathbb{R}^n$ equipped with the usual invariant metric $g$.
Let $U_A\in\mathbb{R}^{n\times r}$ and $...
- 1,495
9
votes
1
answer
529
views
Ricci Curvature on Grassmannian
Suppose $G_r(n)$ is the Grassmannian, which is the collection of all $r$ dimensional subspace in $\mathbb{R}^{n}$ equipped with the usual invariant metric. Let $Ricc(G_r(n))$ be the Ricci curvature ...
- 1,495
5
votes
1
answer
495
views
Volume comparison on Grassmannian
Let $G(r,n-r)$ be the Grassmannian, which can be identified with the space of all rank $r$ projection matrices in $\mathbb{R}^n$. Let $\mu$ be the uniform measure on $G(r,n-r)$. For any $\lambda_1,......
- 1,495
4
votes
3
answers
3k
views
Covariant derivative of determinant of the metric tensor
Let $(M,g)$ be a Riemannian manifold and $g$ the Riemannian metric in coordinates $g=g_{\alpha \beta}dx^{\alpha} \otimes dx^{\beta}$, where $x^{i}$ are local coordinates on $M$. Denote by $g^{\alpha \...
- 131
3
votes
0
answers
139
views
Bishop-Gromov inequality strengthened for anisotropic metrics?
The Bishop-Gromov inequality provides an upper-bound on the rate of growth of volume of a ball of radius $r$ in spaces that have a lower-bound on the Ricci curvature, $Ric \geq (n-1)K$.
(I am ...
- 273
2
votes
1
answer
210
views
$L^{2}$ Betti number
Let $\tilde{X}$ be a non-compact oriented, Riemannian manifold adimits a smooth metric $\tilde{g}$ on which a discrete group $\Gamma$ of orientation-preserving isometrics acts freely so that the ...
- 61
1
vote
0
answers
342
views
What is the meaning of Conjugate radius and Injectivity radius?
I review the text book of differential geometry and I find that the conjugate radius and injectivity radius are still enigmatic for me. Here is a quetion which I confuse it. I don't think it is true, ...
- 1,799
7
votes
2
answers
434
views
Convexity in co-ordinate charts of geodesic balls
Let $g_{ij}$ be a Riemannian metric tensor on an open subset $U\subseteq \mathbb{R}^n$, and let $p\in U$.
I would guess the following is true:
for $\epsilon$ sufficiently small, the $g$-geodesic ...
- 3,212
11
votes
1
answer
592
views
How to construct a nice homotopy?
Let $(M,g)$ be a closed simply-connected Riemannian manifold. Can we find a constant $C$, which depends on $M$, such that for any closed curve $\alpha_0$ with $L(\alpha_0) \le 1$, there exists a ...
- 2,535
22
votes
1
answer
1k
views
Is the metric completion of a Riemannian manifold always a geodesic space?
A length space is a metric space $X$, where the distance between two points is the infimum of the lengths of curves joining them. The length of a curve $c: [0,1] \rightarrow X$ is the sup of
$$ d(c(0),...
- 27.5k
22
votes
15
answers
7k
views
Geodesics on the sphere
In a few days I will be giving a talk to (smart) high-school students on a topic which includes a brief overview on the notions of curvature and of gedesic lines. As an example, I will discuss flight ...
- 3,749
5
votes
1
answer
372
views
Quantitative upper bound on mean curvature of an isometric embedding
By Nash embedding theorem, any complete Riemannian manifold $M$ can be isometrically embedded in $\mathbb{R}^N$, for sufficiently large $N$.
The proof of the theorem is quite involved, and it is not ...
- 3,223
6
votes
0
answers
367
views
Adjoint of the Hodge de Rham star operator under the integral pairing
Given a Riemannian manifold $(M, g)$ of dimension $n$, the Hodge star operator $\star: \Omega^k(M) \to \Omega^{n-k}(M)$ is defined. What is the (formal) adjoint of $\star$ under the integration ...
- 5,824
6
votes
1
answer
449
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 ...
- 690
3
votes
0
answers
77
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 ...
- 5,405
6
votes
2
answers
412
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 ...
- 1,799
7
votes
3
answers
419
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$ ...
- 229
21
votes
6
answers
3k
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$ ...
7
votes
0
answers
305
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 ...
- 6,001
17
votes
3
answers
972
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:...
- 437
1
vote
0
answers
96
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)$$
Define $ G_\sigma: \mathbb{...
- 1,100
1
vote
0
answers
922
views
On a Riemannian manifold, calculate the metric from the distance [closed]
Given a Riemannian manifold $(M,g)$ is it possible to calculate the distance between two points on this manifold. Is it possible the inverse? That means: given a formula of the distance, for example:
...
- 399
8
votes
0
answers
194
views
Geometric mean of three or more positive definite matrices
The geometric mean of two positive definite (Hermitian) matrices of same size is defined by $$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^...
- 13.4k
3
votes
0
answers
150
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 ...
- 13.5k
5
votes
1
answer
375
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 $$\forall x \in M, \...
- 356
5
votes
1
answer
206
views
When can the metric be reconstructed (up to scaling) from knowing the conjugate points?
Let $M$ be a smooth manifold of dimension $\geq 2$. Let $g$ be a complete Riemannian metric on $M$. Let $C \subseteq M \times M$ be the set of pairs of $g$-conjugate points.
The set $C$ doesn't ...
- 63.9k
3
votes
1
answer
237
views
Smoothness of a curve vs. smoothness of the squared distance from the curve to points on Riemann manifolds
I know that the squared distance function from a point $p$ on a Riemann manifold $M$ is smooth in a n-hood of $p$. Therefore for a smooth curve $c:\mathbb{R}\to M$ the concatenation $d(p,\cdot)^2\circ ...
- 41
3
votes
1
answer
143
views
Interpolation inequality related to the 5/3-Laplace operator
I'm having trouble with an estimate that would be helpful in information geometry.
The background is the following. Suppose we have a smooth positive function $g:X \to \mathbb{R}^+$ where $X$ is a ...
- 6,001
1
vote
0
answers
229
views
Distance between quadric surface and point or Intersection of sphere and quadric surface
I asked a similar question on math.stackexchange, but the answer wasn't quite ideal for my application. Apparently analytic solutions are surprisingly rare for general quadric distances.
Given a ...
- 11
31
votes
6
answers
2k
views
If a triangle can be displaced without distortion, must the surface have constant curvature?
Suppose $S$ is a Riemannian 2-manifold (e.g. a surface in $\mathbb{R}^3$).
Let $T$ be a geodesic triangle on $S$: a triangle whose edges are geodesics.
If $T$ can be moved around arbitrarily on $S$ ...
- 151k
14
votes
4
answers
963
views
Steiner's inequality reference request
I remember seeing somewhere that for every connected compact set $\Omega$ in $\mathbb{R}^2$ with piecewise $C^1$ boundary we have
$$A(\Omega_r)\leq A(\Omega)+L(\partial \Omega)r+ \pi r^2,$$
where
$$\...
- 295
7
votes
1
answer
223
views
Five-dimensional manifolds fibering over a fixed hyperbolic surface
I am aware of the classical work by Smale and Barden computing the diffeomorphism type of smooth simply connected 5-manifolds in D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965),...
- 2,630