TITLE: Space of metrics with positive sectional curvature
QUESTION [5 upvotes]: 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.

REPLY [18 votes]: 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.