Positively curved manifolds with large spherical rank

Abstract

Rigidity results are obtained for Riemannian $d$-manifolds with $\sec \ge 1$ 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 $d\ne 6$, for Riemannian manifolds satisfying the Rakić duality principle, and for Kählerian manifolds.