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.

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References
  • Cao, Jianguo; Shaw, Mei-Chi. "A new proof of the Cheeger-Gromoll soul conjecture and the Takeuchi theorem" (PDF). Archived from the original (PDF) on 2004-02-20.
  • Cheeger, Jeff; Gromoll, Detlef (1972), "On the structure of complete manifolds of nonnegative curvature", Annals of Mathematics, Second Series, 96 (3): 413–443, doi:10.2307/1970819, ISSN 0003-486X, JSTOR 1970819, MR 0309010
  • Gromoll, Detlef; Meyer, Wolfgang (1969), "On complete open manifolds of positive curvature", Annals of Mathematics, Second Series, 90 (1): 75–90, doi:10.2307/1970682, ISSN 0003-486X, JSTOR 1970682, MR 0247590
  • Perelman, Grigori (1994), "Proof of the soul conjecture of Cheeger and Gromoll", Journal of Differential Geometry, 40 (1): 209–212, doi:10.4310/jdg/1214455292, ISSN 0022-040X, MR 1285534, Zbl 0818.53056
  • Sharafutdinov, V. A. (1979), "Convex sets in a manifold of nonnegative curvature", Mathematical Notes, 26 (1): 556–560, doi:10.1007/BF01140282
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