The topology of closed manifolds with quasi-constant sectional curvature

TL;DR

This paper classifies closed manifolds with metrics of locally quasi-constant sectional curvature, showing they are topologically composed of space forms and spherical bundles, under certain conditions.

Contribution

It extends the understanding of manifolds with quasi-constant curvature by characterizing their topology and diffeomorphism types under broad conditions.

Findings

Manifolds with generic quasi-constant curvature metrics are graphs of space forms.

Under positivity and torsion-free fundamental group assumptions, they are connected sums of spherical space forms.

These manifolds are diffeomorphic to spherical bundles over the circle.

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.