All Questions
Tagged with reference-request riemannian-geometry
320 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
28
votes
7
answers
11k
views
Roadmap to learning about Ricci Flow?
Hello,
I'm curious to what books etc. one could use to understand the basics of Ricci flow, what areas of math are needed and so? What areas should one specialize in? See it as a roadmap to ...
- 289
28
votes
2
answers
5k
views
The Origin of the Musical Isomorphisms
In Riemannian geometry, the "lowering indices" operator is denoted by $\flat:TM \to T^*M$ and the "raising indices" operator by $\sharp:T^*M \to TM$. These isomorphisms are ...
- 1,202
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
21
votes
2
answers
3k
views
Alternative definitions of Sobolev spaces on non-compact Riemannian manifolds
SHORT VERSION: Does the Meyers-Serrin theorem hold on complete, non-compact Riemannian manifolds, i.e. $W^{k,p}(M) = H^{k,p}(M)$? My guess is that this holds for the special case $k=1$ (and all $p\geq ...
- 3,223
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
17
votes
3
answers
4k
views
Convex hull on a Riemannian manifold
Let $M$ be a complete Riemannian 2-manifold.
Define a subset $C$ of $M$ to be convex
if all shortest paths between any two points
$x,y \in C$ are completely contained within $C$.
For a finite set of ...
- 151k
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
3
answers
6k
views
Linear/Non-linear sigma model
This is slightly an open-ended invitation to discuss references and reasons for excitement about the linear and non-linear sigma model.
I gauge from some other interactions that it has considerable ...
- 3,541
13
votes
3
answers
986
views
A conformal map whose Jacobian vanishes at a point is constant?
Let $f:M \to N$ be a smooth weakly conformal map between connected $d$-dimensional Riemannian manifolds, i.e. $f$ satisfies $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}_{TM}$.
Assume $d \ge 3$ ...
- 6,741
13
votes
2
answers
816
views
$C^0$ estimate for solutions of elliptic PDE with Neumann BC
I am interested in a reference for (or counterexample to) a particular
$C^0$ estimate for solutions of the Laplace equation with Neumann
boundary conditions. More precisely, let $(M,g)$ be a $C^\...
- 131
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
13
votes
1
answer
739
views
Is this expression for the Laplacian of conformal maps between Riemannian manifolds known?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\Hom}{\operatorname{Hom}}$
$\newcommand{\tr}{\operatorname{tr}}$
$\newcommand{\TM}{\operatorname{T\M}}$
$\newcommand{\TN}{\...
- 6,741
12
votes
3
answers
930
views
Voronoi cells and the dual complexes in Riemannian manifolds
I would like to use some "intuitively clear" properties of Voronoi cells in general Riemannian manifolds, but I have trouble finding references.
Let $(X,d)$ be a connected Riemannian ...
- 3,897
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
1k
views
A Riemannian metric on $S^2 \times S^2$ of nonnegative curvature that is not a product
Good afternoon,
There is an example of a Riemannian metric on $S^2 \times S^2$ of nonnegative sectional curvature that is not a product metric. I know there is one; however, I cannot find a specific ...
- 113
11
votes
4
answers
2k
views
Elliptic regularity on compact manifold without boundary
Let $(M,g)$ be a Riemannian compact manifold without boundary, and $\Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this:
For any $u\in H^1(M)$, ...
- 571
11
votes
4
answers
649
views
Introductory textbook on geometry of hyperbolic space
I am looking for an introductory textbook to the geometry of the hyperbolic space $\mathbb{H}^n$. The book should include explicit description of geodesics and horospheres in various models (...
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
1k
views
Who first proved that a vanishing Riemann tensor is sufficient for the existence of Euclidean coordinates?
Riemann famously introduced the notion of what we now call a Riemannian manifold and introduced the Riemann curvature tensor $R_{ijk}{}^l$, showing that it is an obstruction the local existence of ...
- 21.5k
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
2
answers
1k
views
Information about Milnor conjecture
I'm a student of mathematics and I need know about the status of the Milnor conjecture (if there are partial results or if someone solved that). The statement is:
A complete Riemannian manifold with ...
- 203
10
votes
4
answers
667
views
Reference for shortest educational path to (Riemannian) hyperbolic plane
I am teaching an undergraduate class for math majors on axiomatic geometry, culminating in the proof that hyperbolic geometry is equiconsistent* with Euclidean geometry. I would like to make an end-of-...
- 2,218
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
2
answers
2k
views
Infinite dimensional Riemannian geometry
My current research has brought me into an area the requires me to learn some infinite dimensional Riemannian and Kähler geometry. Can someone recommend some good books or survey articles to help me ...
- 283
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
3
answers
2k
views
What spaces have well known horofunctions?
Following Gromov, take a metric space $(X,d)$ and consider $C(X)/\mathbb{R}$ the set of continuous functions to $\mathbb{R}$ with the topology of uniform convergence on compact sets after taking the ...
- 4,304
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
1
answer
1k
views
Bochner formula in different forms
I am looking for a reference (better a book) that contain integral Bochner formulas for domains with boundary (I need it for 1-forms and functions only).
For example I will need the following formula:...
- 45k
10
votes
0
answers
464
views
Reference for sets of locally finite perimeter on Riemannian manifolds
I am looking for a reasonably complete reference for Ennio De Giorgi's theory of sets of locally finite perimeter (also christened by him as Caccioppoli sets, after Renato Caccioppoli's pioneering ...
- 7,817
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