A conformal characterization of manifolds of constant sectional curvature

A special case of the main result states that a complete $1$-connected Riemannian manifold $(M^n,g)$ is isometric to one of the models ${\mathbb{R}}^n$, $S^n(c)$, ${\mathbb{H}}^n(-c)$ of constant curvature if and only if every $p\in M^n$ is a non-degenerate maximum of a germ of smooth functions whose Riemannian gradient is a conformal vector field.