Soul theorem

In mathematics, the soul theorem is a classical theorem of Riemannian geometry. It can be stated as follows:

Related Topics:
Mathematics - Riemannian geometry

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:If (M,g) is a complete non-compact Riemannian manifold with sectional curvature Kge 0, then (M,g) has a compact totally convex, totally geodesic submanifold S such that M is diffeomorphic to the normal bundle of S.

Related Topics:
Complete - Compact - Riemannian manifold - Sectional curvature - Totally convex - Submanifold - Normal bundle

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The submanifold S above is called a soul of (M, g); it is not uniquely determined, but any two souls are isometric.

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The theorem was proved by Jeff Cheeger and Detlef Gromoll, as a generalization of an earlier result of Gromoll and Wolfgang Meyer.

Related Topics:
Jeff Cheeger - Detlef Gromoll - Wolfgang Meyer

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