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][2] and [here][3]).

"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][1]).

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).

[1]:https://math.stackexchange.com/a/1745846/104576
[2]:https://mathoverflow.net/questions/54434/when-can-a-connection-induce-a-riemannian-metric-for-which-it-is-the-levi-civita
[3]:https://math.stackexchange.com/a/517070/104576