Let M be a complete n-dimensional Riemanian manifold with nonnegative Ricci
curvature in which one of the Sobolev inequalities (integral \ f \(p) dv)
(1/p) less than or equal to C (integral \del f \(q) dv) (1/q), f epsilon C-
0(infinity) (M), 1 less than or equal to q < n, 1/p = 1/q - 1/n, is satisfi
ed with C the optimal constant of this inequality in R-n. Then M is isometr
ic to R-n.