All Questions
188 questions
2
votes
0
answers
67
views
Closed-form expression for Riemannian exponential maps on symmetric spaces
Besides the Poincaré model for the hyperbolic disc, $S^n$, and Euclidean space, what are known instances of a symmetric space $M$ (finite-dimensional) for which the exponential map is known in closed-...
- 5,405
1
vote
1
answer
94
views
Initial value problems on manifolds around submanifolds (reference)
I am looking for a reference on initial value problems formulated on smooth manifolds with initial conditions on submanifolds. More precisely, let $X$ be a smooth manifold and $Y\subset X$ a embedded ...
- 11
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
6
votes
2
answers
317
views
Quasi-isometric embedding of graphs in non-compact riemannian surfaces
Given a complete riemannian surface $(S,m)$, where $S$ is homeomorphic to $\mathbb{R}^2$, I would like to find a weighted graph $G$ (which means a graph with real non-negative weights on the edges), ...
- 935
3
votes
0
answers
188
views
References and results for the eigenvalues of Ricci tensor
I am looking for references or results that gives estimates for every eigenvalue of the Ricci tensor. For example, the least eigenvalue is related to the minimum of the Ricci curvature, what can we ...
- 1,774
4
votes
1
answer
301
views
Injectivity radius of parallel hypersurfaces
Let $(M,g)$ be a Riemannian manifold and let $N$ be a compact hypersurface isometrically embedded into $M$ and let $\eta$ denote a choice of unit normal vector field on $N$. It is then true that $N$ ...
- 409
4
votes
0
answers
880
views
Scalar curvature in terms of second fundamental form, reference request
I would like to cite a reference for the following formula for scalar curvature:
If $\Sigma$ is a hypersurface in Euclidean space, then $R=H^2-\lvert A\rvert^2$, where $R$ is the scalar curvature ...
6
votes
3
answers
1k
views
The isometry group of a product of two Riemannian manifolds
Under what conditions is the isometry group of a product of two Riemannian manifolds the product of the isometry groups of each one of the components?
One counterexample is a product of two isometric ...
- 2,535
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
7
votes
1
answer
456
views
Nash embedding theorem for manifolds with boundary
A celebrated theorem of Nash is that every $C^k$ ($k\geq 3$) Riemannian manifold $(M,g)$ can be isometrically embedded into some Euclidean space $\mathbb{R}^d$ for some $d\in \mathbb{N}$. However, I ...
- 409
1
vote
1
answer
331
views
Can divergence free vector fields be approximated by smooth ones?
If $M$ is a compact Riemannian manifold, is the space of $C^{\infty}$ divergence-free vector fields dense in the space of $C^r$ divergence-free vector fields, in the $C^r$ topology ($r\geq 1$)? How ...
- 19
7
votes
2
answers
396
views
Is every metric uniformly close to a metric with negative scalar curvature?
Let $M$ be a smooth manifold with non-empty boundary.
Let $g$ be a smooth Riemannian metric on $M$. Is the following true?
For every $\epsilon >0$ there exist a Riemannian metric $g_{\epsilon}$ ...
- 6,741
2
votes
0
answers
134
views
Hypersurfaces whose unit normal $N$ satisfies $[N,X] =0$ for every tangent vector field $X$
Let $M$ be a hypersurface of a Riemannian manifold, and assume that $M$ satisfies the following property:
For each $p \in M$, given a unit normal vector field $N$ defined in a neighborhood $U$ of $...
- 985
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
1
vote
0
answers
159
views
The Laplacian of a tubular neighborhood
Let $(M,g_M)$ be a compact submanifold of $\mathbb{R}^n$. Are there any known results relating the spectrum of the Laplace-Beltrami operator of M to the spectrum of the Laplace-Beltrami operator of a ...
- 409
6
votes
0
answers
122
views
Given the Ricci decays rapidly to 0 at infinity, is the metric asymptotically flat?
Consider the manifold $M=\mathbb{R}^3 \setminus B$ where B is the ball with radius 1. Let $f \in C^{ \infty}(M) $ satisfying:
$$f = \frac{C(\theta, \phi)}{r} + O( r^{-2}) $$
Where $(r,\theta,\phi)$ ...
- 969
3
votes
0
answers
336
views
Understanding Calabi's conjecture proof: What is it meant by the logarithm of a differential form?
I'm reading several books and articles concerning Yau's proof of the Calabi conjecture. I want to have a deep understading of how and why such proof actually works, but most articles are aimed at ...
- 153
6
votes
2
answers
706
views
Reference request: uniformization theorem
I would appreciate if someone could point me to some introductory literature/resources where I can learn about Poincaré's uniformization theorem at a basic level.
Any good powerpoint notes, short ...
- 63
5
votes
1
answer
594
views
Existence of nonvanishing Killing field
Let $(M,g)$ be a closed Riemannian manifold.
Q Is there any research about the existence of nonvanishing Killing field, especially the nontrivial example.
- 1,915
1
vote
2
answers
1k
views
Reference on Complex Geometry
For the preparation of a complex geometry lecture I am looking for a good literature. I already have standard literature like Huybrechts "Complex Geometry. An Introduction" and I am also using it. But ...
- 39
3
votes
1
answer
628
views
Local Sobolev embedding on complete Riemannian manifold
Let $(M,g)$ be a complete Riemannian $m$-manifold, with bounded geometry and $m\geq2$. Suppose $Ric\geq(n-1)\kappa$. Let $B_p(r)$ be a geodesic open ball.
Q Can we find a constant $C=C(\kappa,r,m)$(...
- 1,915
5
votes
0
answers
307
views
Gradient estimate for Poisson equation on manifold
In Gilbarg-Trudinger's book 'Elliptic Partial Differential Equations of Second order', the maximum principle is used to derive the following gradient estimates for Poisson equations on Euclidean ...
- 2,789
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
2
votes
0
answers
66
views
One-parameter group of nonvanishing vector field
Let $M$ be a smooth manifold( if necessary one can assume it is closed), $V$ be a non-vanishing vector field of $TM$.
Q: Under what condition, we can say that $\overline{\exp(tV)}$, i.e. the closure ...
- 1,915
6
votes
0
answers
355
views
Higher order variations of Riemannian geodesics
Consider a mapping $\Gamma$ from the Euclidean plane or an open subset to a Riemannian manifold $M$ so that each $\Gamma(s,\cdot)$ is a geodesic.
There is a well established theory of the first order ...
- 8,086
4
votes
0
answers
116
views
$\ell_p$ geodesic distance on smooth Riemannian manifold and Logarithmic Sobolev Inequalities
Bear with me, I'm not a professional geometer. Recently, I've been studying Logarithmic Sobolev Inequalities (LSI) for probability distributions on manifolds (e.g as done in works of Bobkvo et al. ...
- 6,853
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
2
votes
1
answer
239
views
Projection of a ball in the ambient space to a manifold
Let $B_h (x)$ be the ball of radius $0<h \ll 1$ centered at $x\in \mathbb{R}^d$.
Let $I=[0,1]^{d-1}$ be the unit cube in $\mathbb{R}^{d-1}$, and let $f:I \to \mathbb{R}$ be a $C^2$ function. Then $...
- 3,574
6
votes
1
answer
399
views
A possible generalization of the exponential map
Let $M$ be a $n$-dimensional Riemannian Manifold, fix $p\in M$, and $1<k<n$. Do we know if the following is true?
For any $k$-dimensional subspace $V$ of $T_p M$, there exists a minimal ...
- 1,630
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
5
votes
1
answer
295
views
Existence of geodesic convex functions
By a result of Shing-Tung Yau [1974, Mathematische Annalen 207: 269-270], there are no non-trivial continuous geodesic convex functions on complete manifolds with finite volume.
What happened if we ...
- 1,991
2
votes
0
answers
195
views
Reference for connection of a Hessian metric
Let $(M,\langle \cdot,\cdot\rangle)$ be a pseudo-Riemannian manifold and $f: M \to \Bbb R$ be a smooth function. One can consider the covariant Hessian $\nabla ({\rm d}f)$. Some time ago I had seen a ...
- 1,163
3
votes
0
answers
348
views
The uniqueness of Poincaré metric
The Poincaré metric $ds=\frac{\sqrt{dx^2+dy^2}}{y}$ has the proprety that the action of the group $PSL(2,\mathbb{R})=SL(2,\mathbb{R})/\{\pm I_{2}\}$ on $\mathbb{H}$ preserves the hyperbolic distance.
...
- 31
1
vote
0
answers
272
views
A cohomology associated to a Riemannian manifold
Let $N$ be a compact Riemanian manifold and $G$ be its isometry group. Let $M=\chi^{\infty}(N)$ be the space of smooth vector fields on $N$. There is a natural right action of $G$ on $M$ with $X.g=g^*(...
- 356
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
3
votes
0
answers
112
views
Is the square root of curl^2-1/2 a natural (Dirac-)operator?
In current computations on a particular $3$-dimensional Riemannian manifold, a first order differential operator $D:\Gamma^\infty(TM,M)\to \Gamma^\infty(TM,M)$ acting on vector fiels shows up, with ...
- 1,942
1
vote
1
answer
291
views
What is the status of the smooth version of bellows conjecture
Bellows conjecture for polyhedra was setteled in 1997. How about the smooth version of it, ie bending of closed 2D submanifolds in $\mathbb{R}^3$ while preserving the Riemannian structure/intrinsic ...
- 1,117
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
4
votes
0
answers
343
views
Riemannian metrics on a manifold with corners
For a smooth manifold with corners (although maybe there is no universally agreed definition of it), is there always a Riemannian metric making every face totally geodesic?
Is there any reference ...
- 803
1
vote
0
answers
162
views
Gromov-Hausdorff relative compactness without curvature restrictions
A famous theorem of Gromov says that the set of compact Riemannian manifolds with $Ric \geq c$ and $\text{diam} \leq D$ is relatively compact in the Gromov-Hausdorff metric. Chapter 10 of the book by ...
- 1,407
3
votes
0
answers
74
views
Deforming a non-positively curved Riemannian manifold into a negatively curved one
Cheeger deformations can be used to deform some non-negatively curved Riemannian manifolds into positively curved manifolds (e.g., sectional curvatures strictly positve), see
What is a Cheeger ...
- 1,942
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
3
votes
1
answer
704
views
Bishop-Gromov volume comparison on manifolds with negligible negative Ricci curvature
Let us consider a complete Riemannian manifold $M$ of dimension $n$ with $Ric \geq 0$. Then the Bishop-Gromov volume comparison theorem says that for any $p \in M$, the function
$$ \frac{\text{Vol}(B(...
- 31
1
vote
0
answers
133
views
Reference for example of gradient steady Ricci solitons
Recently I read a paper about Ricci solitons. I quote a paragraph of it here:
In dimension three, the classification of complete gradient steady Ricci solitons is still open. Known examples are ...
- 4,195
1
vote
0
answers
156
views
Proof of Berger in "Sur les variétés d’Einstein compactes"
I would appreciate any reference that contains either a translation or proof of the following interesting observation of Berger (Sur les variétés d’Einstein compactes, M. Berger paper (in French)).
...
- 4,195
4
votes
2
answers
1k
views
When a Killing vector field on Riemannian manifold $(M,g)$ is gradient?
Let $(M^n,g)$ be a Riemannian manifold that admit a unit Killing vector field $X$. i.e., $\mathscr{L}_Xg=0$. Is it possible that there exist a smooth function $f$ on $M$ such that $X=\mathrm{grad}f$?
...
- 4,195
8
votes
1
answer
400
views
Multidimensional gluing theorem for Riemannian manifolds
I would like to understand whether the following multidimensional (partial) generalization of the A.D. Alexandrov gluing theorem is true and, if yes, whether there is a reference.
(The original ...
- 21.8k
22
votes
2
answers
1k
views
Why doesn't this construction of the tangent space work for non-Riemannian metric manifolds?
In the 1957 paper, On the differentiability of isometries, Richard S. Palais gives a way to construct the tangent spaces of a Riemannian manifold using only its metric space structure (Theorem, p.1).
...
- 2,680
1
vote
1
answer
220
views
Question on $\alpha-$Einstein manifolds
A Riemannian manifold $(M,g)$ is called $\alpha-$Einstein if there exist a non-zero $1-$form $\alpha$ such
$$\rho=ag+b\alpha\otimes\alpha$$
where $a,b$ are smooth functions on $M$ and $\rho$ is ricci ...
- 4,195
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