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:
Such a submanifold is called a soul of.
The soul is not uniquely determined by in general, but any two souls of 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 to be Euclidean space. The sectional curvature is everywhere, and any point of can serve as a soul of.
Now take the paraboloid, with the metric being the ordinary Euclidean distance coming from the embedding of the paraboloid in Euclidean space. Here the sectional curvature is positive everywhere, though not constant. The origin is a soul of. Not every point of is a soul of, since there may be geodesic loops based at, in which case wouldn't be totally convex.
One can also consider an infinite cylinder, again with the induced Euclidean metric. The sectional curvature is everywhere. Any "horizontal" circle with fixed is a soul of. 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:
Grigori Perelman proved this statement by establishing that in the general case, Sharafutdinov's retraction is a submersion. Cao and Shaw later provided a different proof that avoids Perelman's flat strip theorem.