2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61373We show that the indices of certain twisted Dirac operators vanish on a $Spin$-manifold $M$ of positive sectional curvature if the symmetry rank of $M$ is $\geq 2$ or if the symmetry rank is one and $M$ is two connected. We also give examples of simply connected manifolds of positive Ricci curvature which do not admit a metric of positive sectional curvature and positive symmetry rank.21 pages, revised version, new titleDifferential Geometry53C20, 57R20 (Primary) 53C21, 19L47, 53C27 (Secondary)text