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.



Space of metrics with positive sectional curvature
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.


**On the other hand,** since you are interested in metrics with positive sectional curvature at one point. If write the metric is written in the normal coordinates at this point, then the space of germs of Riemannian metrics (at this point) forms a cone.
(By the way, the same holds for negative curvature.)


