Cohn-Vossen's theorem

In mathematics , Cohn-Vossen's theorem is a proposition from the differential geometry of surfaces .

It says that every complete , open , nonnegative curved , 2-dimensional Riemannian manifold is either diffeomorphic to or flat . ${\displaystyle \mathbb {R} ^{2}}$

It is not known whether the analogue of the Cohn-Vossen Theorem holds for manifolds of higher dimensions. A weaker result, which applies in any dimension, is Gromoll-Meyer's theorem .

literature

• Stephan Cohn-Vossen : Shortest paths and total curvature on surfaces , Comp. Math. 2, 69-133, 1935.