Gromoll-Meyer theorem

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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 \R^n.

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

The Bonnet-Myers theorem states that if there is a positive lower bound to the sectional curvature of a Riemannian manifold, then it is compact, and in fact, its universal cover is compact. This in particular implies that any open Riemannian manifold which has everywhere positive sectional curvature, must allow its curvature to take arbitrarily small values.

Conversely, if a manifold is such that the sectional curvature is everywhere positive, but its infimum is zero, then the manifold must be open (by the fact that the image of a map from a compact space to reals attains its extrema). Thus, the Gromoll-Meyer theorem is really about complete Riemannian manifolds whose sectional curvature attains arbitrarily low positive values but is nowhere zero.

Cartan-Hadamard theorem

Further information: Cartan-Hadamard theorem

The Cartan-Hadamard theorem states that a nonpositively curved Riemannian manifold must have a universal cover diffeomorphic to \R^n. Equivalently, if the manifold is itself simply connected, it is diffeomorphic to \R^n.

The Gromoll-Meyer theorem and the Cartan-Hadamard theorem are thus both about diffeomorphism with \R^n, but both have very different flavours: the Gromoll-Meyer assumes everywhere positive sectional curvature, while Cartan-Hadamard assumes everywhere nonpositive sectional curvature. The additional assumptions are also different. While in Gromoll-Meyer, we assume the manifold to be complete and open, Cartan-Hadamard requires the manifold to be simply connected.