Abstract: We prove that if is a compact Riemannian -manifold and if , where , , and are the sectional curvature and Ricci curvature of respectively, and , then is diffeomorphic to a spherical space form. In particular, if is a compact simply connected manifold with and , then is diffeomorphic to the standard -sphere . We also extend the differentiable sphere theorem above to submanifolds in Riemannian manifolds with codimension .
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