Rigidity results are obtained for Riemannian d-manifolds with sec> 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¤ 6, for Riemannian
manifolds satisfying the Rakić duality principle, and for Kählerian manifolds.