The sphere $ S^n $ equipped with a metric of constant positive curvature $ g $ has orientation preserving isometry group $$ Iso^+(S^n,g) \cong SO_{n+1}(\mathbb{R}) $$
Indeed every compact group is the isometry group of some manifold see
However it is not clear to me how this works in particular examples.
Is there a "well-known" family of Riemannian manifolds $ M^n $ whose group of orientation preserving isometries is $ SU_n $?
I know that $ \mathbb{C}P^{n-1} $ with the Fubini Study metric has orientation preserving isometry group $ PU_n $, when $ n $ is even, (see What is the compact Riemannian manifold $M$ such that $SU(n)/\mathbb{Z}_n$ is the isometry group of $M$?) so that's close but not quite it.
