4
votes
3answers
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
2answers
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
votes
1answer
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
0answers
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
2answers
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 ...
9
votes
2answers
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
1answer
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
1answer
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 ...
11
votes
1answer
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
2answers
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
1answer
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 ...
6
votes
2answers
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
1answer
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 ...