  Let M be a complete Riemannian manifold whose sectional curvature is bounded
above by 1. We say that M has positive spherical rank if along every geodesic
one hits a conjugate point at t=\pi. The following theorem is then proved: If M
is a complete, simply connected Riemannian manifold with upper curvature bound
1 and positive spherical rank, then M is isometric to a compact, rank one
symmetric space (CROSS) i.e., isometric to a sphere, complex projective space,
quaternionic projective space or to the Cayley plane.
  The notion of spherical rank is analogous to the notions of Euclidean rank
and hyperbolic rank studied by several people (see references). The main
theorem is proved in two steps: first we show that M is a so called Blaschke
manifold with extremal injectivity radius (equal to diameter). Then we prove
that such M is isometric to a CROSS.
