Gromoll-Meyer theorem
This result is valid in all dimensions
This article describes a result related to the sectional curvature of a Riemannian manifold
Definition
Any complete open positively curved Riemannian manifold is diffeomorphic to .
By positively curved we mean that the sectional curvature is everywhere strictly positive.
Relation with other results
Weakening positivity to nonnegativity
The starting point that inspired the Gromoll-Meyer theorem was the Cohn-Vossen theorem, which states that in dimension 2, nonnegative sectional curvature (which is not everywhere zero) is sufficient to imply that the manifold is diffeomorphic to Euclidean space.
The correct generalization of the Cohn-Vossen theorem to higher dimensions, which would imply the Gromoll-Meyer theorem, is the Cheeger-Gromoll conjecture, which looks at quasi-positively curved Riemannian manifolds as a generalization of two-dimensional nonnegatively curved Riemannian manifolds.
The crux of the argument given by Gromoll and Meyer, which involves the use of simple points, does not, however, easily generalize to a proof of the conjecture.
Bonnet-Myers theorem
Further information: Bonnet-Myers theorem