12
votes
2answers
887 views

Does there exist a closed manifold that can be given both a Euclidean and a Hyperbolic structure?

I originally asked this on math.stackexchange, where I asked if there could exist a closed manifold that could be given different geometric structures of constant curvature (not at the same time, of ...
5
votes
1answer
192 views

Can an open manifold with positive Ricci curvature be non simply connected at infinity?

The question is in the title, I haven't been able to locate a discussion of these kind of properties.
11
votes
3answers
253 views

Are there quanitative versions of Thurston's geometrization for manifolds which fiber over $S^1$?

The geometrization theorem tells us: Theorem (Thurston) The mapping torus $M_\phi$ of a pseudo-Anosov diffeomorphism $\phi: S_g \rightarrow S_g$ from a genus $g$ surface to itself admits a ...
0
votes
1answer
146 views

discrete subgroups of the isometries of a product

Suppose $X_1$ and $X_2$ are two nice metric spaces, e.g. two Riemannian manifolds, and let $G_i=Isom(X_i)$. Then $G_1\times G_2\subset Isom(X_1\times X_2)$. Suppose $X_1\times X_2$ is not compact and ...
8
votes
3answers
306 views

Voronoi cells and the dual complexes in Riemannian manifolds

I would like to use some "intuitively clear" properties of Voronoi cells in general Riemannian manifolds, but I have trouble finding references. Let $(X,d)$ be a connected Riemannian manifold and ...
6
votes
1answer
603 views

How metric is Riemannian geometry

Let $(M, g)$ be a finite-dimensional Riemannian manifold. It is well-known, that the Riemannian metric induce a metric on the manifold by $$d(x, y) = \text{inf} \int_a^b \| \dot\gamma(t) \| \, ...
9
votes
2answers
221 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 ...
18
votes
1answer
613 views

Is the following a sufficient condition for asphericity?

I recently came across the following question while working on some problems on manifolds with lower Ricci curvature bounds. Given $n$ does there exist a large $R>0$ with the following property: ...
7
votes
1answer
488 views

Riemannian metrics on non-paracompact manifolds

After proving the existence of Riemannian metrics on manifolds, one of the students asked if the "paracompactness" is necessary. Of course the standard proof with the partition of unity uses this ...
20
votes
2answers
1k views

Is there a Chern-Gauss-Bonnet theorem for orbifolds?

There's a Gauss-Bonnet theorem for compact 2-orbifolds(due to Satake, I think), which gives a relation between the curvature of a Riemannian orbifold and the orbifold topology(i.e. taking into account ...
6
votes
2answers
770 views

G-spaces and manifolds

In his book "The geometry of geodesics" H. Busemann defines the notion of a G-space to be a space which satisfies the following axioms: The space is metric The space is finitely compact, i.e., a ...
5
votes
1answer
700 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 ...
28
votes
0answers
1k views

Minimal volume of 4-manifolds

This question came up in a talk of Dieter Kotschick yesterday. The minimal volume of a manifold is the infimum of volumes of Riemannian metrics on the manifold with sectional curvatures bounded in ...