Understanding the structure of Riemannian manifolds is an important issue in differential geometry. Many interesting results about relations between curvature and topology of manifolds have appeared during the past years. In this talk, we'll introduce some results about complete open manifolds supporting a Sobolev type inequality. We'll see that complete open manifolds with nonnegative Ricci curvature on which some Sobolev type inequality holds are not very far from the Euclidean space and that Hadamard manifolds support many important inequalities.
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