Welcome to MathOverflow

Question

4

1

Hello;

We know that the space of riemannian metrics on a compact manifold is an open cone in the space of symmetric 2-tensors. Is it reasonable to think that metrics with positive sectional curvature (even positive at a specific point $x \in M$) also form a convex cone?

This is a question about the local behaviour of metrics, so I am not imposing the condition that the sectional curvature be positive everywhere.

Also, due to certain corollary of this statement for a class of metrics, I am quite certain that this cannot hold for metrics of negative curvature.

edited Sep 29 2011 at 0:34

asked Sep 28 2011 at 23:54

S.A.A
237●7

Do you want to impose to have a "convex" cone? otherwise this seems obvious. – Reimundo Heluani Sep 29 2011 at 0:30

Yes Reimundo, the question is whether it is convex or not. – S.A.A Sep 29 2011 at 0:34

I added the word convex to the question. Thanks. – S.A.A Sep 29 2011 at 0:35

1

The following reference does not answer your question, but seems relevant: front.math.ucdavis.edu/0907.2444 There Fernando Coda Marques proves that the space of positive scalar curvature metrics on a given orientable 3-dimensional manifold is path-connected. It makes essential use of the Ricci flow and the work of Perelman. – Benoît Kloeckner Sep 29 2011 at 8:26

Anton Petrunin · Accepted Answer · 2011-09-29 01:44:38Z UTC

No, the formula for curvature is nonlinear with respect to metric tensor in a very essential way.

In particular, a convex combination of two positively curved metrics can have negative curvature. In fact, arbitrary large negative sectional curvature.

For example, the induced metric on any embedding $\mathbb{S}^2\hookrightarrow\mathbb{S}^2\times \mathbb{S}^2$, such that both projections are diffeomorphisms is a convex combination of metrics with constant curvature 1/2. But it is not hard to make such metric arbitrary bad.

Space of metrics with positive sectional curvature

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.