Let $M$ be a smooth manifold. Let $g_1,g_2$ be two rigid Riemannian metrics on it. (i.e with no isometries except the identity).
Is it true that every convex combination of $g_1,g_2$ is also rigid?
I suspect the answer is negative for some $M$. (i.e there are manifolds whose space of rigid metrics is not convex). However, I only have an idea which I am not sure how to turn into a rigorous proof:
Idea: Show there exists two metrics $g_1,g_2$ and a non-rigid metric $h$ such that $h+(g_1-g_2),h+(g_2-g_1)$ are rigid metrics (positivity is not trivial).
Then,
$[ h+(g_1-g_2)]+[h+(g_2-g_1)]=2h$ is a sum of two rigid metrics which is not rigid.
I am not sure how to show this for any particular $M$. Perhaps there is some argument which uses somehow the fact that rigidity of a metric is a "generic" property.