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Concept
Soul theorem
Summary
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. Jeff Cheeger and Detlef Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Wolfgang Meyer. The related soul conjecture, formulated by Cheeger and Gromoll at that time, was proved twenty years later by Grigori Perelman.
Cheeger and Gromoll's soul theorem states:
If (M, g) is a complete connected Riemannian manifold with nonnegative sectional curvature, then there exists a closed totally convex, totally geodesic embedded submanifold whose normal bundle is diffeomorphic to M.
Such a submanifold is called a soul of (M, g). By the Gauss equation and total geodesicity, the induced Riemannian metric on the soul automatically has nonnegative sectional curvature. Gromoll and Meyer had earlier studied the case of positive sectional curvature, where they showed that a soul is given by a single point, and hence that M is diffeomorphic to Euclidean space.
Very simple examples, as below, show that the soul is not uniquely determined by (M, g) in general. However, Vladimir Sharafutdinov constructed a 1-Lipschitz retraction from M to any of its souls, thereby showing that any two souls are isometric. This mapping is known as the Sharafutdinov's retraction.
Cheeger and Gromoll also posed a converse question of whether there is a complete Riemannian metric of nonnegative sectional curvature on the total space of any vector bundle over closed manifolds of positive sectional curvature.
Examples.
As can be directly seen from the definition, every compact manifold is its own soul. For this reason, 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.
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