**19**

votes

**2**answers

2k 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 ...

**16**

votes

**8**answers

1k 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 ...

**38**

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 ...

**4**

votes

**3**answers

472 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 ...

**23**

votes

**5**answers

2k 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 ...

**18**

votes

**0**answers

1k views

**13**

votes

**2**answers

699 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 ...

**6**

votes

**2**answers

489 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

304 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 ...

**18**

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

**1**answer

496 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

208 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 ...

**6**

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 ...

**3**

votes

**2**answers

434 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 ...

**2**

votes

**0**answers

102 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 ...

**3**

votes

**1**answer

642 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

212 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

215 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 ...

**5**

votes

**1**answer

264 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

639 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 ...

**0**

votes

**1**answer

71 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

274 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?