Manifolds tightly covered by two metric balls

Abstract

In this note we provide natural optimal geometric conditions for a Riemannian manifold suitably covered by two open metric balls to be homeomorphic to a sphere. This can be viewed as a geometric analogue of Brown's theorem in topology stating that a closed manifold covered by two topological balls is a sphere.

arXiv commentWith the help of Karsten Grove, the submission is rewirtten. Add two theorems and more connections with previous literatures