Soul theorem

In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Cheeger and Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Wolfgang Meyer. The related soul conjecture was formulated by Gromoll and Cheeger in 1972 and proved by Grigori Perelman in 1994 with an astonishingly concise proof.

The soul theorem states:

If (M, g) is a complete connected Riemannian manifold with sectional curvature K ≥ 0, then there exists a compact totally convex, totally geodesic submanifold S whose normal bundle is diffeomorphic to M.

(Note that the sectional curvature must be non-negative everywhere, but it does not have to be constant.) Such a submanifold S is called a soul of (M, g).

The soul is not uniquely determined by (M, g) in general, but any two souls of (M, g) are isometric. This was proven by Sharafutdinov using Sharafutdinov's retraction in 1979.

Examples

Every compact manifold is its own soul. Indeed, the theorem is often stated only for non-compact manifolds.

As a very simple example, take M to be Euclidean space Rⁿ. The sectional curvature is 0 everywhere, and any point of M can serve as a soul of M.

Now take the paraboloid M = {(x, y, z) : z = x² + y²}, with the metric g being the ordinary Euclidean distance coming from the embedding of the paraboloid in Euclidean space R³. Here the sectional curvature is positive everywhere, though not constant. The origin (0, 0, 0) is a soul of M. Not every point x of M is a soul of M, since there may be geodesic loops based at x, in which case wouldn't be totally convex.

One can also consider an infinite cylinder M = {(x, y, z) : x² + y² = 1}, again with the induced Euclidean metric. The sectional curvature is 0 everywhere. Any "horizontal" circle {(x, y, z) : x² + y² = 1} with fixed z is a soul of M. Non-horizontal cross sections of the cylinder are not souls since they are neither totally convex nor totally geodesic.

Soul conjecture

Cheeger and Gromoll's soul conjecture states:

Suppose (M, g) is complete, connected and non-compact with sectional curvature K ≥ 0, and there exists a point in M where the sectional curvature (in all sectional directions) is strictly positive. Then the soul of M is a point; equivalently M is diffeomorphic to Rⁿ.

Grigori Perelman proved this statement by establishing that in the general case K ≥ 0, Sharafutdinov's retraction P : M → S is a submersion. Cao and Shaw later provided a different proof that avoids Perelman's flat strip theorem.

References