Soul theorem

History

The soul theorem was proved by Cheeger and Gromoll in their paper On the structure of complete manifolds of nonnegative curvature.

Statement

Any open Riemannian manifold $(M,g)$ with nonnegative sectional curvature contains a compact totally geodesic submanifold $S$ (called the soul), such that $M$ is diffeomorphic to the normal bundle on $S$, viz $\nu (S)$.

References

• On the structure of complete manifolds of nonnegative curvature by Jeff Cheeger and Detlef Gromoll, The Annals of Mathematics, 2nd Ser., Vol. 96, No. 3 (Nov., 1972), pp. 413-443
• Improving the metric in an open manifold with nonnegative curvature by Luis Guijarro, Proceedings of the American Mathematical Society, Volume 126, Number 5, May 1998, Pages 1541-1545

External links

• Improving the metric in an open manifold with nonnegative curvature