Cheeger-Gromoll conjecture

Statement

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

Progress towards the conjecture

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 is replaced by quasi-positivity.

References

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