So this is my question :
Let $M$ be a smooth manifold. With any riemaniann metric $g$ on $M$ comes an isometry group $I(g)$. Intuition (well at least mine, which may be flawed...) suggests that there should be a metric $g$, not necessarily unique, that gives maximal symmetry to $M$ in the following sense :
if $g'$ is any riemannian metric on $M$ then there is an injective morphism from $I(g')$ to $I(g)$.
The example I have in mind is the sphere $S^2$ : it seems clear that the usual metric gives maximal symmetry and that any other metric would either give the same group of symmetry or one strictly smaller.
Does anyone know if this is true or not ? If so can you give me the argument or a reference ?