  A result of M. Ledoux is that a complete Riemannian manifold with non
negative Ricci curvature satisfying the Euclidean Sobolev inequality is the
Euclidean space. We present a shortcut of the proof. We also give a refinement
of a result of B-L. Chen et X-P. Zhu about locally conformally flat manifolds
with non negative Ricci curvature. Eventually, we discuss Ledoux's result when
the hypothesis on the Ricci curvature is weakened on a hypothesis on the scalar
curvature.
