Cheeger-Gromoll conjecture

This article describes a result related to the sectional curvature of a Riemannian manifold

History

Proposal as a conjecture

The conjecture was made by Cheeger and Gromoll in their celebrated paper On the structure of complete manifolds of nonnegative curvature.

Proof

The conjecture was proved by work of Perelman, following from Perelman rigidity theorem.

Statement

Any complete open quasi-positively curved Riemannian manifold is diffeomorphic to $\mathbb {R} ^{n}$.

Relation with other results

Cohn-Vossen theorem

The conjecture is known to be true in dimension two. This is the content of the Cohn-Vossen theorem.

Gromoll-Meyer theorem

This proves a weaker form of the conjecture where positivity replaces quasi-positivity.

References

• On the structure of complete manifolds of nonnegative curvature by Jeff Cheeger and Detlef Gromoll