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:If (''M'',''g'') is a Complete ''non''- Compact Riemannian Manifold with Sectional Curvature , then (''M'',''g'') has a Compact Totally Convex , Totally Geodesic Submanifold ''S'' such that ''M'' is diffeomorphic to the Normal Bundle of ''S''. The submanifold ''S'' above is called a soul of (''M'', ''g''); it is not uniquely determined, but any two souls are isometric. The theorem was proved by Jeff Cheeger and Detlef Gromoll , as a generalization of an earlier result of Gromoll and Wolfgang Meyer . SOUL CONJECTURE In the same paper Cheeger and Gromoll gave the following conjecture: :Suppose ''M'' is complete and noncompact with sectional curvature , but at some point. Then soul of ''M'' has to be a point (or equivalently ''M'' is diffeomorphic to ). The conjecture was open for about 20 years, and was solved by Grigori Perelman with a surprisingly short argument. REFERENCES
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