All Questions
188 questions
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 ...
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
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
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
18
votes
5
answers
4k
views
What are good Morse Theory lecture notes and books?
Searching on the net I couldnt find any recent lecture/course notes on Morse Theory. I found an old set of notes (http://www.math.toronto.edu/mgualt/Morse%20Theory/mfp.pdf) by Mike Hutchings and these ...
- 2,246
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
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
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
16
votes
5
answers
2k
views
Reference request: Recovering a Riemannian metric from the distance function
Let $M = (M, g)$ be a Riemannian manifold, and let $p \in M$.
Writing $d$ for the geodesic distance in $M$, there is a function
$$
d(-, p)^2 : M \to \mathbb{R}.
$$
This function is smooth near $p$. ...
- 27.7k
15
votes
6
answers
2k
views
Any shortcuts to understanding the properties of the Riemannian manifolds which are used in the books on algebraic topology
I'm now attending a reading seminar on the algebraic topology.
The seminar treats the book of Bott & Tu (Differential Forms in Algebraic Topology) and Milnor (Characteristic Classes).
In those ...
- 1,013
15
votes
2
answers
2k
views
Riemannian manifold as a metric space
I am looking for a reference to the following simple statement; it must be classical. (It is easy to proof, but I want to have a reference.)
A metric space $X$ that corresponds to a Riemannian ...
- 45k
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
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
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
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
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
13
votes
2
answers
2k
views
Is there a solution of the Yamabe problem using Ricci flow?
Someone told me that it is possible to solve the Yamabe problem using Ricci flow. The proof I know of is the one originally proposed by Yamabe and then completed by Trudinger, Aubin and Schoen (in ...
- 5,146
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
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
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
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
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
11
votes
2
answers
1k
views
Non-compact manifolds of positive/non-negative Ricci curvature
Consider a non-compact complete Riemannian manifold $(M, g)$ with smooth compact boundary $\partial M$. Suppose also that $M \setminus \partial M$ has positive/non-negative Ricci curvature.
My ...
- 111
11
votes
1
answer
529
views
Length decreasing homotopies of curves
Let $M$ be smooth compact riemannian manifold with boundary and $\varphi_0: S^1\to M$ be a rectifiable curve (or a smooth one). I would like to find a reference to the following statement:
Statement. ...
- 14.3k
10
votes
1
answer
3k
views
Taylor expansion of the metric tensor in the normal coordinates
I am looking for a reference with a Taylor expansion of the metric tensor in the normal coordinates.
The coefficients should be written in terms of $\mathrm{Rm}, \nabla\mathrm{Rm}, \nabla^2\mathrm{Rm},...
- 45k
10
votes
1
answer
707
views
Injectivity radius of manifolds with boundary
This question stems from the discussion in:
how to define the injectivity radius of manifolds with boundary?
Suppose $(M,g)$ is a compact Riemannian manifold with boundary. In this context, let ...
- 409
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
10
votes
1
answer
403
views
Positive Ricci curvature on fiber bundles
My advisor and I are working on Ricci curvature and an anonymous referee pointed out the following conjecture:
Let $F\hookrightarrow M\stackrel{\pi}{\to}B$ be a fiber bundle from a compact manifold ...
- 1,774
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
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 ...
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
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
9
votes
1
answer
2k
views
Is there a book on differential geometry that doesn't mention the notion of charts?
What are some books/texts that use chart free coordinate free language for things otherwise written in a coordinate based formulation? I would like to learn about covariant differentiation, curvature, ...
- 319
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
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
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
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
8
votes
1
answer
856
views
Are there mistakes in Kovalev's "Twisted connected sums and special Riemannian holonomy"?
This is kind of a strange and vague question... sorry about that.
I am really interested in $G_2$ Twisted Connected sums as described in this paper: https://arxiv.org/abs/math/0012189 "Twisted ...
user117832
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
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
1
answer
230
views
The closure of the space of Riemannian metrics with a fixed isometry class
Let $M$ be a closed manifold, and let $\mathscr{M}$ be the space of all Riemannian metrics over $M$. It is known that this is a Fréchet manifold. Consider also $\mathscr{D}$ the diffeomorphisms group ...
- 808
8
votes
1
answer
375
views
Harmonic functions on complete Riemannian manifolds
I have started reading a paper of Colding and Minicozzi, where they prove that on a complete Riemannian manifold $M$ of non-negative Ricci curvature, the space of harmonic functions of growth order at ...
- 81
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
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
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
8
votes
1
answer
218
views
Existence of properly discontinuous and cocompact action
Let $M$ be a complete Riemannian manifold. My question is, under what conditions does $M$ admit a discrete group of isometries $\Gamma$ which acts properly discontinuously and cocompactly on $M$, that ...
- 81
8
votes
1
answer
673
views
Classification of compact globally symmetric spaces
It is known that any connected compact Lie group $G$ is a finite quotient of the product of a compact simply connected semisimple Lie group $\tilde{G}$ and a torus $\mathbb{T}^n$ (see for example ...
- 193
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
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
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