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.
The conjecture was proved by work of Perelman, following from Perelman rigidity theorem.
Relation with other results
The conjecture is known to be true in dimension two. This is the content of the Cohn-Vossen theorem.
This proves a weaker form of the conjecture where positivity replaces quasi-positivity.
- On the structure of complete manifolds of nonnegative curvature by Jeff Cheeger and Detlef Gromoll