  Rigidity results are obtained for Riemannian $d$-manifolds with $\sec
\geqslant 1$ and spherical rank at least $d-2>0$. Conjecturally, all such
manifolds are locally isometric to a round sphere or complex projective space
with the (symmetric) Fubini--Study metric. This conjecture is verified in all
odd dimensions, for metrics on $d$-spheres when $d \neq 6$, for Riemannian
manifolds satisfying the Raki\'c duality principle, and for K\"ahlerian
manifolds.
