Soul theorem

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

Soul conjecture

In the same paper Cheeger and Gromoll gave the following conjecture:

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:Suppose, M is complete and noncompact with sectional curvature Kge 0, but K > 0 at some point. Then soul of M has to be a point (or equivalently M is diffeomorphic to {mathbb R}^n).

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The conjecture was open for about 20 years, and was solved by Grigori Perelman with a surprisingly short argument.

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