Complete manifolds of non-negative sectional curvature largely reduce to the compact case
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.
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.[2]
Very simple examples, as below, show that the soul is not uniquely determined by (M, g) in general. However, Vladimir Sharafutdinov constructed a
1-Lipschitzretraction from M to any of its souls, thereby showing that any two souls are
isometric. This mapping is known as the
Sharafutdinov's retraction.[3]
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.[4]
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 spaceRn. The sectional curvature is 0 everywhere, and any point of M can serve as a soul of M.
Now take the
paraboloidM = {(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.[5]
One can also consider an infinite
cylinderM = {(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.[6]
Soul conjecture
As mentioned above, Gromoll and Meyer proved that if g has positive sectional curvature then the soul is a point. Cheeger and Gromoll conjectured that this would hold even if g had nonnegative sectional curvature, with positivity only required of all sectional curvatures at a single point.[7] This soul conjecture was proved by
Grigori Perelman, who established the more powerful fact that Sharafutdinov's retraction is a
Riemannian submersion, and even a
submetry.[8]
Sharafutdinov, V. A. (1979). "Convex sets in a manifold of nonnegative curvature". Mathematical Notes. 26 (1): 556–560.
doi:
10.1007/BF01140282.
S2CID119764156.