3
votes
0answers
169 views

What is known about analogous results of Kazdan and Warner in higher dimensions?

First let me state a Theorem due to Kazdan and Warner: ``Let M be a compact two dimensional orientable manifold. Let $f: M \rightarrow \mathbb{R}$ be a function that has the same sign as ...
10
votes
2answers
508 views

A riemannian manifold with finitely many closed contractible geodesics

By a closed geodesic, I mean a smooth periodic geodesic $\mathbb{R} \rightarrow (M,g)$. I will consider them up to geometric distinction. This means that any two closed geodesics are equivalent if ...
1
vote
1answer
402 views

Heisenberg group: research themes

I am currently studying the Heisenberg group from the Riemannian geometry point of view, particularly focusing on its Gromov boundary and more generally its metric properties. I would like to know ...
13
votes
1answer
405 views

Are isospectral manifolds necessarily homeomorphic?

It's known that there are pairs of closed Riemannian manifolds which are isospectral but not isometric. Is it known if there are closed Riemannian manifolds which are isospectral but not ...
10
votes
0answers
330 views

Best metrics on exotic R^4

What is known about the existence of complete metrics with good properties (e.g., Einstein, constant scalar curvature, etc...) on exotic ${\bf R}^4$s? Note, that some exotic ${\bf R}^4$s have ...
4
votes
2answers
397 views

Higher derivatives than Jacobi fields.

Hi, the first and second derivatives of the distance function (either the full $d:M\times M\to \mathbb{R}$ function or the $d(p,\cdot):M\to \mathbb{R}$ function) as well as the derivative of the ...
6
votes
2answers
648 views

Inverse function theorem on manifolds with boundary

I wonder that whether there exists a version of the inverse function theorem for smooth maps from a smooth manifolds with boundary to a smooth manifold without bounary? More precisely, whether the ...
0
votes
1answer
252 views

Numbers associated with boundaries of manifolds

I don't know what name if any is attached to the numbers I'm about to describe. For a line segment, [a,b] the number is 1 if for any k in (a,b) and 2 if k=a or k=b. For a square, [a,b] ...
4
votes
2answers
267 views

volume growth of tubular neigbhorhood of critical values of an algebraic/differentiable map

Edit: One can also assume $M,N$ compact in the following, and that $M$ is equipped with a Riemannian metric as well. Incorporated Mike's suggestion. Sard's theorem says that for every smooth map $f: ...
3
votes
4answers
761 views

Does every smooth manifold of infinite topological type admit a complete Riemannian metric?

To elaborate a bit, I should say that the question of the existence of a complete metric is only of interest in the case of manifolds of infinite topological type; if a manifold is compact, any metric ...
1
vote
2answers
1k views

Are all Riemannian metrics induced by Euclidean metrics? [Nash Embedding Theorem]

Let $M$ be a smooth manifold. We can get a Riemannian metric on $M$ by at least two methods: first by partitions of unity and second by the Whitney embedding theorem: we can embed $M$ into a ...