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SPHERICAL POINTS IN RIEMANNIAN MANIFOLDS

A point p in a Riemannian manifold M is weakly spherical if for each point q ≠ p there is either exactly one or at least three minimizing geodesic segments joining p to q. In this note, it is shown that round 2-dimensional spheres are the only Riemannian surfaces with a weakly spherical point realizing the injectivity radius.