All Questions
188 questions
4
votes
1
answer
139
views
Convex hull of a connected subset on a complete surface of non-positive curvature
Let $S$ be a simply connected surface, possibly with boundary components, with a smooth complete metric of non-positive curvature. Let $X\subset S$ be a closed connected subset. I would like to know ...
- 14.3k
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
2
votes
0
answers
305
views
"Riemannian" collar theorem
Let $(M,g)$ be a compact manifold of dimension $n$ with boundary. If $\partial M$ is smooth then one has a control on the determinant of the Jacobian of the diffeomorphism in the collar theorem, i.e....
- 143
8
votes
1
answer
795
views
Reverse Toponogov triangle comparison
See the wiki page https://en.wikipedia.org/wiki/Toponogov%27s_theorem
One consequence of the Toponogov comparison Theorem is that if the sectional curvature of a manifold $M$ is pinched below by a ...
- 319
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
3
answers
613
views
Manifolds with special holonomy especially $G_2$
I am interested in learning about $G_2$ manifolds and am aware that one of the canonical references is Joyce's Compact Manifolds with Special Holonomy. I am certain that my background is, at this ...
- 21
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
4
votes
1
answer
565
views
Riccati equation and principal curvatures
Let $\Omega$ be an open subset of a Riemannian manifold $M$. Assume that $\Sigma:=\partial \Omega$ is $C^2$.
Let $U$ be a neighborhood of $\Omega$ such that $\exp_p(t\nu(p))$ is diffieomorphism, ...
- 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
8
votes
4
answers
710
views
Torsion of submanifolds
Studying curves in the Euclidean three dimensional space, one usually defines the curvature and the torsion of a curve. If I am not missunderstanding the thing, I guess that a curve has zero torision ...
- 2,384
4
votes
1
answer
347
views
Some questions on a paper of Wilking
I am currently trying to understand Wilking's paper "A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities" (DOI: 10.1515/crelle.2012.018, arXiv:1011....
- 81
3
votes
2
answers
1k
views
Reference for homogeneous spaces
I am a graduate student of differential geometry.
I would like to get an overview over the way, how results are usually obtained for homogeneous spaces by Lie algebraic methods. By definition a ...
- 79
2
votes
3
answers
397
views
Reference request for structure equations
Let $(M,g)$ be a Riemannian manifold and let $\lbrace e_1,...,e_n\rbrace$ be a locally frame field on $M$ and $\omega _1 ,...,\omega _n$ be the dual $1$-forms of it. If $\omega _{ij}$ be the ...
- 601
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
1
answer
496
views
Ricci flow preserves holonomy
Could someone please give me a reference where I can find a complete proof of the result Ricci flow preserves holonomy? Is there any way to prove that Ricci flow preserves Kahler condition without ...
- 789
13
votes
4
answers
3k
views
General Relativity and Differential Geometry intuitions of Second Bianchi Identity
In General Relativity, one uses the Riemann Tensor in its coordinate form $R_{abcd}$, and proves the Second Bianchi Identity-
$R_{abcd;e} + R_{abde;c} + R_{abec;d} = 0$
It is said that ...
- 3,574
12
votes
1
answer
3k
views
how to define the injectivity radius of manifolds with boundary?
For manifolds without boundary one defines the injectivity radius as the maximal radius where the exponential map is a diffeomorphism. One can then show that the injectivity radius is the maximum ...
- 687
4
votes
2
answers
219
views
Is $\mathbb{P}T^*M$ a sub-Riemannian manifold if $M$ is Riemannian?
(this question is about a particular aspect of a previous question, which was not duly stressed)
Let $(M,g)$ a Riemannian $n$-dimensional manifold, and let
$$
\widetilde{M}:=\mathbb{P}T^*M
$$
be the $...
- 2,527
5
votes
1
answer
1k
views
On the complexification of a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle. If we suppose $TM\otimes\mathbb{C}$ is the complexification of $TM$ then how can we define a natural metric on the complex bundle $...
- 601
1
vote
0
answers
82
views
Scattering in (pseudo-)Riemannian spaces
I will ask my question in a broad way, leaving a lot of freedom for answers.
Suppose that we have a (pseudo-)Riemannian space $(M,g)$ and we fix some ball-like domain $B \subset M$. Suppose you are ...
- 1,461
2
votes
0
answers
232
views
Intuitive understanding of the mean curvature flow [closed]
I am trying to develop some intuition into the properties of mean curvature flow of a surface in $\mathbb{R}^3$. As an example, I am trying to understand what happens to a surface of revolution $S = (...
- 21
5
votes
0
answers
179
views
Some questions on the nodal geometry of Dirac operators
Let me begin by quoting a well-known result of Christian Baer (see here). The result goes as follows:
Theorem (Baer): Consider a connected $n$-dimensional Riemannian manifold with Dirac bundle $S$ ...
- 1,407
4
votes
2
answers
281
views
Heat kernel asymptotics for small distances
I heard a talk where the speaker said that on a Riemannian manifold, for small values of $\text{dist }(x, y)$, the heat kernel $p_t(x, y)$ satisfies
$$p_t(x, y) = \frac{1}{(4\pi t)^{n/2}}e^{-\frac{\...
- 41
3
votes
0
answers
153
views
Gaussian heat kernel bounds on Riemannian manifolds [duplicate]
I wish to know if we have Gaussian lower and upper bounds for the heat kernel,i.e. $$
t^{-n/2}e^{-\frac{\rho(x,y)^2}{C_1t}} \lesssim p_t(x,y) \lesssim t^{-n/2}e^{-\frac{\rho(x,y)^2}{C_2t}},
$$
on a ...
- 31
13
votes
2
answers
789
views
Geometric characterization of martingales
Recently I've read a paraphrasing from Ito saying that he sometimes thinks of martingales as geodesics in a very large dimensional manifold.
My question is, is there any research studying this idea?
...
- 5,405
2
votes
1
answer
262
views
A clarification regarding analytic perturbation of metrics and Laplacian
This question is in reference to the following Mathoverflow question and the accepted answer to it. It seems to me that it is taken for granted that if the metric $g_t$ perturbs real analytically in ...
- 123
3
votes
1
answer
560
views
Prescribing an induced metric
We know that, if we have a surface $z=f(x,y)$ with Euclidean space being ambient manifold, the induced metric is as follows (in matrix form):
$$g=\begin{bmatrix}
1+\left ( \frac{\partial f(x,y)}{\...
- 267
5
votes
2
answers
307
views
Compact surface with arbitrarily large eigenvalue
Consider a compact surface $M$ with genus $\gamma \geq 2$ and fix a positive real number $V$. Is it known whether it is possible to produce a metric $g$ on the surface $M$ such that $(M. g)$ has ...
- 51
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
12
votes
1
answer
1k
views
Multiplicity of Laplace eigenvalues
Disclaimer: This is a very heuristic question and I will be satisfied with heuristic insights, if rigorous and precise answers are not possible.
All the examples of closed surfaces (or higher ...
- 121
13
votes
3
answers
2k
views
Isometry group of a compact hyperbolic surface
Consider a compact surface $M$ of genus $g \geq 2$ with a metric of constant negative curvature. My question is, is it known under what sorts of sufficient conditions such a metric will have non-...
- 133
7
votes
2
answers
725
views
Ricci flow and isometry group
It is known (via Kotschwar's uniqueness of backwards Ricci flows) that the isometry group of a Riemannian metric remains unchanged under the Ricci flow. But, one can easily observe that it can change ...
- 73
9
votes
5
answers
1k
views
List of generic properties of Riemannian metrics
I am highly interested in compiling a list of generic properties of Riemannian metrics on a (may be compact) manifold in general, or under "relatively broad" assumptions, like generic properties of ...
4
votes
0
answers
95
views
Laplacian Spectra on Nearly Nodal Riemann Surfaces
Consider a family of complex curves ${\mathcal C} \to {\mathbb D}$ such that the central fibre is a nodal Riemann surface while other fibres are smooth Riemann surfaces. We choose a family of ...
- 1,207
2
votes
0
answers
123
views
Mean value operator on Riemannian manifold
Let $(M,g)$ a Riemannian manifold. Further $M$ should be a harmonic space, that is $M$ is a symmetric and simply connected space of rank 1. (Example: Spheres $S^n$)
Consider the mean value operator, ...
- 21
5
votes
1
answer
345
views
Convergence of Riemannian metrics spectra
Consider a one-parameter real analytic family of metrics $g_t$ on a compact manifold $M$ converging to a metric $g$ in $C^k$-norm, for some $k$. It is known that the Laplace spectrum of $g_t$ will ...
- 51
13
votes
1
answer
481
views
A question on a result of Colin de Verdière
Consider a compact connected surface $M$ of some genus $\gamma \geq 2$. A particular case of a famous result of Colin de Verdière (see Construction de laplaciens dont une partie finie du spectre est ...
- 1,407
4
votes
0
answers
152
views
Faster (than normal) convergence of the normalized Ricci flow on surfaces
Consider a compact surface $M$ of genus $\gamma > 1$ (I am using the more usual letter "$g$" to denote metric), and the normalized Ricci flow on it. It is known that at time $t$, the scalar ...
- 41
3
votes
3
answers
243
views
Compact surfaces with arbitrary gaps in spectrum
Consider a sequence of positive numbers $a_n$. My question is, can we select a closed Riemann surface whose spectrum $\lambda_i$ satisfies the condition that $\lambda_{i + 1} - \lambda_i > a_i$? Of ...
- 31
10
votes
0
answers
284
views
Comparing spectra of Laplacian and Schrödinger operator
Let $M$ be a closed (compact without boundary) Riemannian manifold. Is there a body of results that compares the eigenvalues of the Laplace-Beltrami operator with that of Schrödinger operators $-\...
- 109
3
votes
1
answer
255
views
Norm on space of metrics
I recently heard a differential geometry talk where the speaker constructed a one-parameter family of metrics $g(t)$ on a smooth manifold and said that $g(t)$ is real analytic in the Banach space $BC(...
- 51
9
votes
1
answer
2k
views
Is a manifold generically real analytic (with generic real analytic metric)?
I have heard it said in some differential geometry talks that "the generic situation in such and such case is real analytic". My question is, is the generic smooth manifold also real analytic in some ...
- 123
5
votes
3
answers
550
views
Can the conformal structure on the projective light-cone detect hyperplane sections?
Let $(V,\langle\,\cdot\,,\,\cdot\,\rangle)$ be an $(n+1)$-dimensional real vector space, equipped with a nondegenerate symmetric bilinear form of indefinite signature, and denote by $\nu(v):=\langle v,...
- 2,527
8
votes
1
answer
421
views
$C^k$ one-parameter family of metrics
Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...
- 1,407
3
votes
1
answer
284
views
Long time existence of Ricci flow on compact surfaces of negative curvature
Is there a long time existence for the Ricci flow on compact negatively curved surfaces? I just read that the normalized Ricci flow has a long time solution converging to a metric of constant negative ...
- 39
4
votes
3
answers
5k
views
Green's function on sphere
Consider radial (normal) coordinates on a sphere $S^n, n \geq 2$. Let the "origin" be the north pole $(0, 0,..., 1)$ and the coordinates be denoted by $(r, \theta)$. We know that the Laplacian $\...
- 41
64
votes
12
answers
22k
views
Advanced Differential Geometry Textbook
I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help.
In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual ...
8
votes
3
answers
2k
views
What does it mean that the Hessian is proportional to the metric?
Let $(M,g)$ be a smooth manifold equipped with a metric tensor $g$, and $f\in C^\infty(M)$ a regular function (i.e., with nowhere vanishing differential).
Denote by $\mathrm{Hess}_g(f):=\nabla df$ ...
- 2,527
18
votes
2
answers
4k
views
Where is the exponential map a diffeomorphism?
Let $M$ be a closed compact Riemannian manifold.
The exponential map $\mathrm{exp}:TM\to M\times M$ takes $(p,v)$ to $(p,\gamma_v(1))$, where $\gamma_v$ is the geodesic flow at $p$ in the direction ...
- 4,588
2
votes
0
answers
168
views
Sources on evolution of submanifolds subject to Ricci flow
I am seeking any textbook or paper addressing the evolution of submanifolds of a manifold undergoing Ricci Flow. Please, any pointer towards this topic is more than welcome.
This old MO post may be ...
- 133