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 , 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 K > 0 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

• 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.