Positively curved manifolds with large spherical rank

TL;DR

This paper proves rigidity results for positively curved manifolds with high spherical rank, confirming conjectures in specific dimensions and classes, and showing such manifolds are locally isometric to standard symmetric spaces.

Contribution

It verifies a conjecture that manifolds with certain curvature and rank conditions are locally isometric to spheres or complex projective spaces in various cases.

Findings

All odd-dimensional cases verified.

Manifolds with the Rakić duality principle confirmed.

Kählerian manifolds satisfy the conjecture.

Abstract

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.