# Tagged Questions

**4**

votes

**3**answers

511 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 ...

**7**

votes

**2**answers

288 views

### Easy proof of topological property of Zoll manifolds

It is known that the cohomology ring of a Zoll manifold---a riemannian manifold all of whose geodesics are periodic with the same minimal period---must be the same as the cohomology ring of a compact ...

**5**

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**1**answer

415 views

### What are the Dirac operators on $S^1$?

This is crossposted at stack exchange as http://math.stackexchange.com/questions/248391/dirac-operators-on-s1.
I am trying to understand the Dirac operators associated to the 2 spinor bundles on ...

**1**

vote

**0**answers

94 views

### About Thom Theorem (representation submanifold for $H_{n-2}(M^n)$)

Recall Thom theorem : If $M^n$ is a smooth orientable closed manifold then any homology class in $H_{n-2}(M)$ is represented by the fundamental class of a smooth submanifold.
And in the Harper and ...

**4**

votes

**2**answers

256 views

### Are negatively pinched manifold locally conformally flat?

One knows that hyperbolic manifolds are locally conformally flat.
How about those negatively pinched manifolds, i.e. the sectional curvature $K$ satisfy:
$$
-\Lambda \le K \le -\lambda$$
for ...

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votes

**2**answers

217 views

### Is compact flat manifold cusp cross-sections of a complete finite volume hyperbolic manifold?

Let $M^{n-1}$ be a closed flat manifold. Is it true that there exists a hyperbolic manifold $N^n$ with finite volume such that $M$ is a cusp cross-section of $N$?
It was proved in "On the geometric ...

**6**

votes

**1**answer

325 views

### Preissmann and Byers Theorems

I'm starting to study at the elementary level the relationship between topology and geometry of a Riemannian manifold of negative curvature. The first two theorems, simple and interesting in this ...

**5**

votes

**1**answer

1k views

### Good Surface,Bad Surface-Surface classification

Maybe this question be very simple, but I don't know why it is hard for me. Thanks for any guide and help.
We say a surface $S$ (2-dimensional metric(compact) Riemannian surface) is good (denote by ...

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votes

**1**answer

628 views

### Analytic Torsion in the Derived Category

I recently learned about analytic torsion and about the amazing Cheeger-Muller theorem identifying analytic and Reidemeister torsion for compact Riemannian manifolds.
Now analytic torsion is defined ...

**7**

votes

**2**answers

688 views

### Euler characteristic, Gauss-Bonnet, and a product formula

I know very little about the Pfaffian or how it works, and I'm new at Riemannian geometry in general. But I was wondering if there is some way to make this "intuitive" argument for the fact that a ...

**1**

vote

**1**answer

887 views

### Betti Numbers (homology vs cohomology)

I'm somewhat confused about the definitions of Betti numbers for Riemannian manifolds. Working with the first Betti number as an example, I have usually taken the definition to be the rank of the ...

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votes

**2**answers

810 views

### For which $b$ it is possible that $S^n$ can have a Lorentz metric? Why?

It is possible that on a sphere $S^n$ there is a natural Riemannian metric in $R^(n+1)$. But it is not always possible for pseudo Riemann metric since the sum of two symmetric matrix which are not ...

**0**

votes

**1**answer

190 views

### altering curvature on a tessellation representation of a compact surface

I have been reading about tessellation representations of compact surfaces, such as how the square tiling the plane represents the torus. For surfaces of genus > 1 (the ones that interest me), we ...