  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.
