Positively curved Killing foliations via deformations

Abstract

We show that a manifold admitting a Killing foliation with positive transverse curvature and maximal defect fibers over finite quotients of spheres or weighted complex projective spaces. This result is obtained by deforming the foliation into a closed one, while maintaining transverse geometric properties, which allows us to apply results from the Riemannian geometry of orbifolds to the space of leaves. We also show that the basic Euler characteristic is preserved by such deformations, which provides us some topological obstructions for Riemannian foliations.