All Questions
188 questions
8
votes
1
answer
1k
views
Spectrum of the Laplacian on p-forms on the sphere
In this paper the authors give an explicit description of the eigenforms and spectrum of the Laplacian acting on $p$-forms on the round sphere $S^n$, apparently citing an unpublished computation of ...
- 585
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
3
votes
1
answer
432
views
Is there a characterization of Riemannian manifolds that split off two factors?
Some Riemannian manifolds are expressed as a product manifold. Recently, I have read two articles about space-times. In both articles, the authors prove that a Riemannian manifold $\bar{M}^n$ is ...
- 422
1
vote
0
answers
224
views
Characterization of the Riemann curvature tensor [duplicate]
Let $(M^n,g)$ be a Riemannian manifold, $a\in M$ be a fixed point. It it well known that there exists a coordinate system near $a$ (e.g. the normal one) such that
$$g_{ij}(x)=\delta_{ij}+O(|x|^2).$$
...
- 21.8k
5
votes
1
answer
482
views
Besse p134 Riemann tensor in dimension 4
Does someone have a reference for the proof of 4.72 page 134 of Einstein Manifolds? It is said that
$$\check{R}-\vert R\vert^2g/4=S/3 (Ric-S/4) +2\mathring{W}(Ric -S/4) $$
because we are in dimension ...
- 914
1
vote
2
answers
139
views
Reference request: minimal (maximal) Lorentzian surfaces in $\mathbb{R}^{1,2}$
Let $R^{1,2}$ be the Minkowski 3-space, I would like to know any references about minimal (maximal) orientable Lorentzian surfaces in $\mathbb{R}^{1,2}$, including examples and maybe general theories, ...
- 783
3
votes
0
answers
256
views
Uniqueness of scalar curvature
I'm reading Gromov's notes
http://www.ihes.fr/~gromov/topics/SpacesandQuestions.pdf
and at page 7 they say that there is a unique second order differential operator $S$ from the space of Riemannian ...
- 585
7
votes
2
answers
2k
views
The integral of torsion
I found the following * exercise (exercise *9) in page 407 of the book of do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced ...
- 356
9
votes
1
answer
509
views
A question on generalized Einstein metrics on four-dimensional manifolds
I am thinking of a possible generalization of Einstein metrics (or a possible characterization of Einstein metrics) on four-dimensional manifolds,
\begin{equation*}
\mathrm{Ric}\circ\mathrm{Ric}=\...
- 399
5
votes
2
answers
704
views
Ricci curvature under rough convergence
From the work of Lott--Villani and Sturm, I know that the following fact holds:
(*) Suppose that $(M_k,g_k,dvol_{g_k})$ is a sequence of compact Riemannian manifolds of non-negative Ricci curvature ...
- 7,197
8
votes
1
answer
336
views
Short examples that are/are not quantum-ergodic
Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic?
Note that a (compact) Riemannian manifold is said to be quantum ergodic if almost ...
- 131
10
votes
1
answer
470
views
Monograph or rich survey on infinite-dimensional Riemann manifolds
I'm working with the space of smooth curves $\mathcal{C}$ in a smooth manifold $M$, having (different, pre-determined) fixed endpoints. I'd like to endow it with a Riemann structure (I already have a ...
- 5,407
18
votes
2
answers
4k
views
Reference request: Geodesic flow on a manifold with negative curvature is ergodic
I'm reading about the Mostow's rigidity theorem, and the proof uses the following (maybe well-known) result:
The geodesic flow on a manifold with negative curvature is ergodic.
The lecture note that ...
- 957
14
votes
1
answer
1k
views
Spectrum of Laplacian in non-compact manifolds
What can be said about the spectrum of the Laplace-Beltrami operator on a non-compact, complete Riemannian manifold of finite volume? For example, is the point spectrum non-empty?
What would be a ...
- 13.5k
4
votes
1
answer
699
views
Spectrum of the Laplace-Beltrami operator on $L^p$: where is it?
On a noncompact Riemannian manifold $M$, the $L^2$-spectrum of the Laplace-Beltrami operator $\Delta$ sits inside $\mathbb{R}$ (by self-adjointness), either to the left or to the right of $0$ ...
user14166
2
votes
1
answer
490
views
Curve on a surface defined by its geodesic curvature
Suppose that $S$ is a smooth complete surface, and $c\colon [0,L]\to S$ is a smooth curve in $S$, parametrized by arc-length. Then $c$ is uniquely determined by its initial tangent vector and its ...
1
vote
0
answers
463
views
Reference request for parallel transport
I am learning about parallel transport on a Riemannian manifold equipped with an affine connexion. It seems (if I understand it well) that, in general, we might not be able to compute the parallel ...
- 119
6
votes
0
answers
352
views
How to generate a random (Weyl) curvature operator ?
Given a dimension $n$, the space of curvature operators is the space $S^2_B(\Lambda^2\mathbb{R}^n)$ of symmetric endomorphisms $R$ of $\Lambda^2\mathbb{R}^n$ which satisfy the first Bianchi identity :
...
- 4,123
0
votes
1
answer
339
views
Polarisation in a neighbourhood of a Lagrangian submanifold
Let $(X, \omega)$ be a symplectic manifold of dimension $2n$ and $\omega$ is an exact symplectic form i.e. $\omega = -d\alpha$. Let furthermore $M \subset X$ be a compact Lagrangian submanifold such ...
- 339
7
votes
1
answer
502
views
Fundamental groups of compact manifolds with non-negative Ricci curvature.
I would like to find an appropriate reference for the following statement:
Statement. Let $M$ be a compact Riemannian manifold with non-negative Ricci curvature.
Then $\pi_1(M)$ is virtually abelian.
...
- 14.3k
7
votes
1
answer
497
views
Open problems about CMC hypersurfaces with symmetries?
Recently, Andrews and Li announced a complete classification of CMC ($H=const.$) tori in $S^3$, confirming a conjecture of Pinkall and Sterling. Their main result is that any such torus is ...
- 5,891
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
0
votes
1
answer
314
views
G-structures and complete riemannian manifolds
what are possible fundamental and introductory texts about G-structures ?
and where i can find the proof of this proposition:
if G(group) acts properly discontinuously on a space X , then G is a ...
- 165
2
votes
1
answer
551
views
Heisenberg group: research themes
I am currently studying the Heisenberg group from the Riemannian geometry point of view, particularly focusing on its Gromov boundary and more generally its metric properties.
I would like to know ...
- 31
7
votes
4
answers
3k
views
Levy-Gromov Isoperimetric Inequality
In his paper "Paul Levy's Isoperimetric Inequality", Gromov gives the following isoperimetric inequality:
Let $V$ be a closed $(n+1)$-dimensional Riemannian Manifold with $\mathrm{Ric}(V) \geq n \...
- 1,123
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
36
votes
10
answers
6k
views
Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature
A curve in the plane is determined, up to orientation-preserving
Euclidean
motions, by its curvature function, $\kappa(s)$.
Here is one of my favorite examples, from
Alfred Gray's book,
Modern ...
- 151k
8
votes
3
answers
1k
views
Higher derivatives than Jacobi fields
The first and second derivatives of the distance function (either the full $d:M\times M\to \mathbb{R}$ function or the $d(p,\cdot):M\to \mathbb{R}$ function) as well as the derivative of the ...
- 516
7
votes
1
answer
1k
views
Helmholtz-Decomposition on compact Riemannian manifolds
For smooth domains $\Omega$ in $\mathbb{R}^n$ it is known that one can decompose vector fields in $L^p(\Omega)^n$, $1 < p <\infty $ into a "gradient"- and a "divergence-free"-part such that
$L^...
- 73
6
votes
1
answer
1k
views
Basic results in bounded geometry
I'm doing analysis (dynamical systems) in the context of Riemannian manifolds of bounded geometry and I find myself reproving quite a few standard results/tools from standard differential geometry, ...
- 2,481
4
votes
2
answers
1k
views
Special Killing Vector Fields
Consider $(M^{n},g)$ to be a Riemannian manifold and suppose that $X$ is a smooth non-trivial Killing vector field on $M$. Away from the zeros of $X$ we have a natural distribution $D$ of $(n-1)$-...
- 2,299
18
votes
2
answers
1k
views
The geometry of Nadirashvili's complete, bounded, negative curvature surface
I would like to understand the geometric structure of
a surface that Nadirashvili constructed which resolved what
was known as Hadamard's Conjecture.
Perhaps in the 15 years since his construction, ...
- 151k
13
votes
4
answers
2k
views
Algebraic surfaces and their (intrinsic) geometry
Recently I began to consider algebraic surfaces, that is, the zero set of a polynomial in 3 (or more variables). My algebraic geometry background is poor, and I'm more used to differential and ...
- 579
2
votes
1
answer
1k
views
Harmonic coordinates on Riemannian manifolds
I'm trying to read the paper of Jost and Karcher on the existence of harmonic coordinates on a ball whose size only depend on the injectivity radius and a two sided bound on the curvature.
...
- 4,123
7
votes
4
answers
3k
views
How does curvature change under perturbations of a Riemannian metric?
Let $M$ be a compact subset of $\mathbb R^2$ with smooth boundary, and let $g$ be a Riemannian metric on $M$. If $g'$ is another Riemannian metric which is "close" to $g$, then they should have ...
- 8,512
6
votes
1
answer
342
views
Contracting a geodesic on a space of curvature less than 1
I would like to ask for a reference to the following statement (hopefully correct):
Let $M$ be a manifold of sectional curvature at most $1$ and let $\gamma$ be a closed geodesic.
Suppose that $\...
- 28.9k
11
votes
2
answers
2k
views
Retraction of a Riemannian manifold with boundary to its cut locus
This question is edited following the comment of Joseph. He pointed out that the main object of the first version of this question is the cut locus.
Recall that the cut locus of a set $S$ in a ...
- 28.9k
9
votes
2
answers
7k
views
Constant curvature manifolds
In two different books I found these two related statements.
The book by Jost defines a ``locally symmetric space" as one for which the curvature tensor is constant and which is geodesically complete....
- 3,541