It is well known fact that Riemannian metric always exist on any manifold. Another way to say that is that any manifold always admit $O(n)$ structure i.e. $O(n)$ reduction of structure group of frame bundle $L(M)$.
One way to see that (next to the classical argument with partition of unity) is that $Gl(n)/O(n)$ is contactible, existance of a $G$ structure is equivalent to the existance of section of a fibre bundle $L(M)/G$ and fibre bundles with contractible fibre always possess one.
My questions are:
$\bullet$ What are another closed subgroups $G \subset Gl(n)$ with $Gl(n)/G$ contractible ?
$\bullet$ What are another closed subgroups $G \subset Gl(n)$ such that any manifold admits $G$ structure?
For compact case, there is essentially one such subgroup up to conjugation, which is $O(n)$. Indeed, one can apply Corollary 1.4 in this paper, which says that for a compact subgroup $H$ in an almost connected locally compact group $G$, $G/H$ is contractible if and only if $H$ is maximal compact. In the case $G = GL(n,\mathbb R)$ they are conjugate of $O(n)$.