Gromoll-Meyer theorem (positive curvature)

The set of Gromoll-Meyer is a theorem from the mathematical field of differential geometry . It was proven by Detlef Gromoll and Wolfgang Meyer .

It says that a complete , positively curved , open manifold is diffeomorphic to Euclidean space . ${\displaystyle \mathbb {R} ^{n}}$

He generalizes Cohn-Vossen's theorem for surfaces, for which even a weaker condition is sufficient: A complete, non-compact surface of non-negative curvature, the curvature of which is positive in at least one point, is diffeomorphic to . It is an open question whether this weaker condition is sufficient in higher dimensions as well. ${\displaystyle \mathbb {R} ^{2}}$

The Bonnet-Myers theorem states that a Riemannian manifold whose intersection curvature has a positive lower bound must be compact. Positively curved, open manifolds necessarily have points at which the sectional curvatures of individual planes come as close as desired to zero.

literature

• Detlef Gromoll, Wolfgang Meyer: On complete open manifolds of positive curvature , Annals of Mathematics 90 (1969), 75-90.