Manifolds tightly covered by two metric balls

TL;DR

This paper establishes optimal geometric conditions under which a Riemannian manifold covered by two metric balls is homeomorphic to a sphere, extending a topological theorem into a geometric context.

Contribution

It provides the first natural geometric criteria for manifolds covered by two metric balls to be topologically spherical, bridging geometry and topology.

Findings

Manifolds covered by two metric balls are homeomorphic to spheres under certain conditions.

The results are optimal and extend Brown's topological theorem to Riemannian geometry.

The paper offers a geometric analogue of a classical topological covering theorem.

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.