**21**

votes

**2**answers

3k views

### Does the curvature determine the metric?

Hello,
I ask myself, whether the curvature determines the metric.
Concretely: Given a compact Riemannian manifold $M$, are there two metrics $g_1$ and $g_2$, which are not everywhere flat, such that ...

**23**

votes

**9**answers

2k 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 ...

**0**

votes

**1**answer

172 views

### elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...

**39**

votes

**7**answers

4k views

### Riemannian surfaces with an explicit distance function?

I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...

**10**

votes

**7**answers

3k 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 ...

**10**

votes

**3**answers

550 views

### Characterizing Hessians among symmetric bilinear tensors

I apologize in advance if this is somewhat elementary, but:
Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\in Sym^2(M)$ are ...

**4**

votes

**3**answers

554 views

### Uniquely geodesic and CAT(0) spaces?

Improvement after J-M Schlenker's comment below :
This post has been divided into two parts, the second part is here.
Question : Is a finite dimensional metric space, uniquely geodesic if and only ...

**24**

votes

**5**answers

3k views

### A geometric interpretation of the Levi-Civita connection?

Let $M$ be a Riemannian manifold. There exists a unique torsion-free connection in the (co)tangent bundle of $M$ such that the metric of $M$ is covariantly constant. This connection is called the ...

**31**

votes

**5**answers

2k views

### Is the Laplacian on a manifold the limit of graph Laplacians?

Here's the sort of thing I have in mind. Let $M$ be a Riemannian manifold, compact if it helps, and let $\Delta_M$ be the Laplace-Beltrami operator. Choose a sequence of triangulations of $M$ so ...

**19**

votes

**0**answers

1k views

**13**

votes

**2**answers

730 views

### Is there a unified reason that there are an infinite number of geodesics between nonconjugate points on a compact manifold?

The proof of this statement seems to break into two really different arguments. So, I'm wondering if there is a better argument that can explain them both, or whether it's really just two theorems ...

**15**

votes

**2**answers

782 views

### If there is a dense geodesic, are almost all geodesics equidistributed? Dense?

Let $M$ be a complete finite volume Riemannian manifold and $\gamma : \mathbb{R}^{\geq 0} \to M$ a geodesic. Suppose that $\mathrm{im}(\gamma)$ is dense. Is it equidistributed in the Riemannian ...

**11**

votes

**1**answer

649 views

### Does the gluing procedure in Robert Wald’s book *General Relativity* yield a Hausdorff spacetime?

Before I state my problem, let me provide some definitions pertaining to the Cauchy Problem in General Relativity.
Definition 1: A triplet $ (\Sigma,h,k) $ is called an initial data set if $ ...

**7**

votes

**1**answer

1k views

### Global description of the Levi-Civita connection

I'm interested in finding a global (coordinate-free) description of the Levi-Civita connection on a (possibly infinite-dimensional) Riemannian manifold X.
I'm not looking for a description of this ...

**6**

votes

**2**answers

520 views

### Ricci curvature of the symplectic group

Is the Ricci curvature of the compact symplectic group $Sp(n)$ bounded below by $cn$ for some constant $c > 0$ independent of $n$?
For $O(n)$ and $U(n)$ I know many references which state such a ...

**5**

votes

**1**answer

332 views

### Proof of the general expression for anomaly in a CFT and its partition function

I think the statement is that for any dimensional CFT the following is true,
$$\langle T^{\mu}_\mu \rangle = \sum B_n I_n - 2(-1)^{d/2}AE_d,$$
where $E_d$ is the `"Euler density" and $I_n$ are ...

**19**

votes

**2**answers

1k views

### Area of distance sphere in manifold with Ricci $\ge 0$.

Let $M$ be a open complete manifold with Ricci curvature $\ge 0$.
By a theorem of Calabi and Yau, the volume growth of $M$ is at least of linear.
I am wondering whether the following statement is ...

**8**

votes

**2**answers

620 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 ...

**8**

votes

**1**answer

526 views

### Surfaces all of whose geodesics are both closed and simple

The Zoll surfaces have the property that all of their geodesics are closed.
If one futher stipulates that all geodesics are also simple, i.e., non-self-intersecting,
does this leave only the sphere?
...

**3**

votes

**1**answer

217 views

### Length spectrum and Zoll surfaces of revolution

The earlier MO question, "Length spectrum of spheres," asked if the length spectrum of closed
geodesics determines the metric on $S^2$, and the answer was a clear No due to Zoll surfaces,
all of whose ...

**12**

votes

**1**answer

284 views

### If all balls around two points are isometric… — manifold version

This question is a natural follow-up of this other question, asked earlier today by wspin.
Let's say that a metric space $(X,d)$ has two poles if:
there are two distinct points $x$, $y$ such that ...

**3**

votes

**2**answers

488 views

### A Converse to the Gauss Bonnet Theorem

Let $S$ be a compact surface in $\mathbb{R}^{3}$ with the gauss normal map $N:S\to \mathbb{S}^{2}$. Assme that $\phi;\mathbb{S}^{2}\to S$ is a diffeomorphism. Put $F=N\circ \phi$ and represent ...

**3**

votes

**2**answers

328 views

### Hypersurfaces and Elliptic Points

I'm reading a paper, in which we have $M^n$ an n-dimensional compact hypersurface embedded in $\mathbb{R}^{n+1}$. We take the scalar cuvature $R$ to be the elementary symmetric polynomial of degree 2 ...

**2**

votes

**0**answers

139 views

### What are Euler density and Weyl invariants?

I would like to know as to what is the definition and significance of what are called "Euler density" and "Weyl invariants" (of weight $-d$ on a $d-$manifold)
Do many (which?) of them vanish when ...

**2**

votes

**2**answers

900 views

### Metric Connections on a Lie Group

A Lie group has three standard Cartan connections; the (-)-connection, the (0)-connection, and the (+)-connection. The (0)-connection is Levi-Civita with the associated metric the bi-invariant metric. ...

**14**

votes

**1**answer

651 views

### Is there any progress on Problem 12 (from Schoen and Yau)?

I saw the following question from the "Problem Section" in Schoen and Yau, page 281, problem 12:
Let $M_1, M_2$ each have negative curvature. If $\pi_1 (M_1)=\pi_1 (M_2)$, prove that $M_1$ is ...

**4**

votes

**2**answers

492 views

### Limit cycles as closed geodesics(geodesible flow)

The classical Van der Pol equation is the following vector field on $\mathbb{R}^{2}$:
\begin{equation}\cases{\dot{x}=y-(x^{3}-x)\\ \dot{y}=-x}\end{equation}
This equation defines a foliation on ...

**3**

votes

**1**answer

674 views

### Orthogonal complements in Hilbert bundles

It's a standard fact that for a finite-dimensional vector bundle with an inner product, the othogonal complement of any subbundle is itself a locally trivial vector bundle.
What is known about the ...

**2**

votes

**1**answer

237 views

### Is there a lower bound for variance in terms of curvature?

If the Gaussian curvature of the metric $g= f^2(x,y)(dx^2+dy^2)$ is nonzero then $f$ cannot be constant. This can be expressed by stating that the (probabilistic) variance $Var(f)$ of $f$ is nonzero ...

**1**

vote

**2**answers

220 views

### Examples on small cut radius of totally convex set in non-negatively curved manifold

Suppose $M^n$ is an open complete nonnegatively curved Riemannian manifold. In Cheeger-Gromoll's proof of the soul theorem. They need an estimate on the cut radius of a totally convex set $C$. By a ...

**0**

votes

**1**answer

237 views

### Why don't $\mathbb{T}^n, \mathbb{S}^n, \mathbb{H}^n$ admit other metrics of constant curvature?

The torus $\mathbb{T}^n$, the sphere $\mathbb{S}^n$ and the hyperbolic space $\mathbb{H}^n$ admit metrics of constant (sectional) curvature $0, 1, -1$ respectively. Do they afford metrics of constant ...

**5**

votes

**1**answer

296 views

### Minimal distance spheres in complex projective spaces

My question has to do with distance spheres in $\mathbb CP^{n+1}$. I am interested in knowing what is the radius $r$ of a distance sphere $S(r)$ around a point that makes it a minimal submanifold ...

**5**

votes

**1**answer

716 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 vesrion of this question is the cut locus.
Recall that the cut locus of a set $S$ in a ...

**1**

vote

**0**answers

441 views

### A metric on $S^{2}$ [closed]

Edit:Can this new version of this question be answered with the method of same comments to the previous version?
Let $p:S^{3}\to S^{2}$ be the Hopf fibration $p(z,w)= (\parallel ...

**1**

vote

**0**answers

118 views

### The “Rolle theorem” for sections of a vector bundle

1)Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...

**0**

votes

**1**answer

79 views

### An special isometric embedding

Let $M$ be a Riemannian manifold and $\gamma$ be a non closed geodesic.
Is there an isometric embedding of $M$ into some $\mathbb{R}^{n}$ which send $\gamma$ into an straight line?
The second ...

**0**

votes

**2**answers

307 views

### Non simply connected HyperKähler 4-manifolds without ALE metrics

In a 1989 paper Peter Kronheimer showed that each simply connected HyperKähler 4-manifold possesses an ALE metric. What do we know about the non-simply connected cases?