Cheeger-Gromoll conjecture
This article describes a result related to the sectional curvature of a Riemannian manifold
Contents
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 .
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