General Sobolev Inequality on Riemannian Manifold
  Let M be a complete n-dimensional Riemannian manifold, if the sobolev inqualities hold on M, then the geodesic ball has maximal volume growth; if the Ricci curvature of M is nonnegative, and one of the general Sobolev inequalities holds on M, then M is diffeomorphic to $R^{n}$. 
Qihua Ruan and Zhihua Chen