The topology of closed manifolds with quasi-constant sectional curvature

Abstract

We prove that closed manifolds admitting a generic metric whose sectional curvature is locally quasi-constant are graphs of space forms. In the more general setting of QC spaces where sets of isotropic points are arbitrary, under suitable positivity assumption and for torsion-free fundamental groups they are still diffeomorphic to connected sums of spherical space forms and spherical bundles over the circle.