I want to get a feeling for how much flexibility we have when putting a Riemannian metric on a given smooth manifold $M$.
Is it always possible to find two non-isometric metrics on $M$? If the answer is positive, is there some qualitative\quantitative estimate on how many different (non-isometric) metrics exists on $M$?
I understand the set of all Rimeannian manifolds might be an infinite dimensional manifold but I think this notion distinguishes between two isometric metrics, while I am asking with "how many" metrics we are left with after we identify those which are isometric.