Does the group of Diffeomorphisms act transitively on the space of Riemannian metrics? Let $M$ be a smooth manifold (maybe compact, if that helps). Denote by $\operatorname{Diff}(M)$ the group of diffeomorphisms $M\to M$ and by $R(M)$ the space of Riemannian metrics on $M$. We obtain a canonical group action
$$ R(M) \times \operatorname{Diff}(M) \to R(M), (g,F) \mapsto F^*g, $$
where $F^*g$ denotes the pullback of $g$ along $F$. Is this action transitive? In other words, is it possible for any two Riemannian metrics $g,h$ on $M$ to find a diffeomorphism $F$ such that $F^*g=h$? Do you know any references for this type of questions?

The group of C^1 diffeomorphism does not act transitively on the space of Riemannian metrics on a compact manifold. For example, two circles of different radius have different diameters and by pullback we can "copy" the metric of one of them on the other. Doing so we get two metrics on a circle with different diameters, hence they can not be C^1 related. 
Thus, the point is that "the diameter" is a C^1-invariant hence an homotety of the metric change the diameter and the "new metric" have different diameter. 
For non compact manifolds, by using a theorem of Nomizu, there exist both complete and non-complete riemannian metrics. See http://www.oberlin.edu/faculty/jcalcut/Nomizu_Ozeki_1961.pdf 
Since "completeness" is a C^1 invariant we get that the group of C^1 diffeomorphism does not act transitively on the space of riemannian metrics.
h. 
