This result relates information on curvature to information on topology of a manifold
This result is valid in all dimensions
Here are some equivalent forms:
- Any complete simply connected Riemannian manifold whose Ricci curvature everywhere is bounded below by a positive constant, must be compact
- Any complete Riemannian manifold whose Ricci curvature everywhere is bounded below by a positive constant, must be compact
- If a Riemannian manifold has the property that there exists a positive constant that lower-bounds the
Any Einstein metric with positive cosmological constant, for instance, satisfies the hypotheses for the Bonnet-Myers theorem, and hence, any manifold possessing such a metric is compact with finite fundamental group.
Relation with other results
Further information: Cartan-Hadamard theorem The Cartan-Hadamard theorem talks of the analogous statement when the manifold has negative sectional curvature throughout. It says that under that assumption, the universal cover is diffeomorphic to real Euclidean space.
Together, the Cartan-Hadamard theorem and Bonnet-Myers theorem tell us that a manifold which has positive curvature bounded from below, cannot be diffeomorphic to a manifold which has negative sectional curvature throughout.