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Boundary convexity on manifolds with nonnegative Ricci curvature

We introduce a new geometric invariant Λ to measure the convexity of the boundary of a riemannian manifold with nonnegative Ricci curvature in the interior. Based on a theorem of Perelman, we are able to show that this new invariant has topological implications. More specifically, we show that if Λ is close to 1 and the sectional curvature is positive on the boundary, then the manifold is contractible.