We show that, with some technical conditions, an Abelian monoidal category admits a monoidal embedding into the category of bimodules over a ring. The case of semisimple rigid monoidal categories is studied in more detail.

In this paper, we use the theory of critical points of distance functions to study the rigidity and topology of Riemannian manifolds with sectional curvature bounded below. We prove that an n-dimensional complete connected Riemannian manifold M with sectional curvature KM [ges ] 1 is isometric to an n-dimensional Euclidean unit sphere if M has conjugate radius bigger than π/2 and contains a geodesic loop of length 2π. We also prove that if M is an n([ges ]3)-dimensional complete connected Riemannian manifold with KM [ges ] 1 and radius bigger than π/2, then any closed connected totally geodesic submanifold of dimension not less than two of M is homeomorphic to a sphere.