Manifolds tightly covered by two metric balls (Q2363177)

A Theorem by \textit{M. Brown} [Bull. Am. Math. Soc. 66, 74--76 (1960; Zbl 0132.20002)] shows that any closed manifold covered by two open Euclidean balls is topologically a sphere. On the other hand, in the Riemannian setting, any closed manifold admits a Riemannian metric such that \(M\) is covered by two open metric balls, i.e., for any \(\epsilon>0\) and \(r>0\), there exists a Riemannian metric on \(M\) so that \(M=B(p,r+\epsilon)\cup B(q,r+\epsilon)\) where \(d(p,q)=2r\), \(B(p,s)=\{x\in M:d(p,x)<s\}\) and \(d(p,x)\) is the Riemannian distance. The author gives a natural optimal geometric condition to ensure that a Riemannian manifold suitably covered by two open metric balls is homeomorphic to a sphere.