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In mathematics, the soul theorem is a classical theorem of Riemannian geometry. It can be stated as follows:
If (M,g) is a complete non-compact Riemannian manifold with sectional curvature <math>K\ge 0<math>,
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 <math>K\ge 0<math>, but <math>K > 0<math> at some point. Then soul of M has to be a point (or equivalently M is diffeomorphic to <math>{\mathbb R}^n<math>).
The conjecture was open for about 20 years, and was solved by Grigori Perelman with a surprisingly short argument.
References
Cheeger, Jeff; Gromoll, Detlef On the structure of complete manifolds of nonnegative curvature. Ann. of Math. (2) 96 (1972), 413--443.
Gromoll, Detlef; Meyer, Wolfgang On complete open manifolds of positive curvature. Ann. of Math. (2) 90 1969 75--90.
Perelman, G. Proof of the soul conjecture of Cheeger and Gromoll. J. Differential Geom. 40 (1994), no. 1, 209--212.
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