  We investigate the topology of the compact submanifolds in round spheres that
satisfy a lower bound on the Ricci curvature depending only on the length of
the mean curvature vector of the immersion. Just in special cases, the limited
strength of the assumption allows some strong additional information on the
extrinsic geometry of the submanifold. This hints that in the more general
situation a full geometric classification may be an unattainable goal.
