Soul conjecture
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:
The 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.
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, 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. 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.
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