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 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 \geq 0

, then there exists a compact totally convex, totally geodesic submanifold

S

such that

M

is diffeomorphic to the normal bundle of

S

.

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.

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 $R n$ . The sectional curvature is $0$ , and any point of $M$ can serve as a soul of $M$ .

Now take the paraboloid $M = {(x, y, z) : z = x 2 + y 2$ }, with the metric $g$ being the ordinary Euclidean distance coming from the embedding of the paraboloid in Euclidean space $R 3$ . 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 $x$ .

One can also consider an infinite cylinder $M = {(x, y, z) : x 2 + y 2 = 1$ }, again with the induced Euclidean metric. The sectional curvature is $0$ everywhere. Any "horizontal" circle ${(x, y, z) : x 2 + y 2 = 1$ } with fixed $z$ is a soul of $M$ .

Soul conjecture

Cheeger and Gromoll's soul conjecture states:

Suppose

(M, g)

is complete, connected and non-compact with sectional curvature

K \geq 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

R n

.

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

References

Cao, Jianguo; Shaw, Mei-Chi. "A new proof of the Cheeger-Gromoll soul conjecture and the Takeuchi theorem" (PDF).
Cheeger, Jeff; Gromoll, Detlef (1972), "On the structure of complete manifolds of nonnegative curvature", Annals of Mathematics. Second Series, 96: 413–443, doi:10.2307/1970819, ISSN 0003-486X, MR 0309010
Gromoll, Detlef; Meyer, Wolfgang (1969), "On complete open manifolds of positive curvature", Annals of Mathematics. Second Series, 90: 75–90, doi:10.2307/1970682, ISSN 0003-486X, MR 0247590
Perelman, Grigori (1994), "Proof of the soul conjecture of Cheeger and Gromoll", Journal of Differential Geometry, 40 (1): 209–212, 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