On manifolds with non-negative Ricci curvature and Sobolev inequalities (Q1293494)

The following result is given. A complete \(n\)-dimensional Riemann manifold \(M\) with nonnegative Ricci curvature is isometric to \(\mathbb{R}^n\) if for all \(C^{\infty}\) compactly supported functions \(f\) on \(M\) the Sobolev inequality \((\int|f|^p dv)^{1/p}\leq C(\int |\bigtriangledown f|^q dv)^{1/q} (1/q)-(1/p)=1/n\) is satisfied with ``optimal'' \(C\).