Let $E$ be a smooth vector bundle over a manifold $M$, equipped with a connection $\nabla$.

The set of $\nabla$-compatible metrics on $E$ forms a convex cone. This cone can be empty, however (see here and here).

"How big" can this cone be? Is it always (or ever) a manifold? (in the finite dimensional setting not every convex cone is a manifold, but closed ones are).

What is its maximal dimension (as a function of $\dim M,\dim E$)? Can this cone be infinite dimensional? non-zero but finite dimensional?

Also, it would be interesting to know what is the minimal non-zero dimension possible (is it greater than one?).

(To summarize, I am asking "which numbers" - including infinity - can be realized as dimensions of this cone).