  We provide a somewhat geometric proof of a rigidity theorem by M. Ledoux and
C. Xia concerning complete manifolds with non-negative Ricci curvature
supporting an Euclidean-type Sobolev inequality with (almost) best Sobolev
constant. Using the same technique we also generalize Ledoux-Xia result to
complete manifolds with asymptotically non-negative curvature.
