Positively curved Killing foliations via deformations

TL;DR

This paper demonstrates that compact manifolds with positively curved Killing foliations can be characterized as fibers over quotients of spheres or weighted projective spaces, using deformation techniques that preserve transverse geometric properties.

Contribution

It introduces a deformation method to analyze Killing foliations with positive transverse curvature, linking their structure to well-understood orbifold geometries and preserving key topological invariants.

Findings

Foliations deform into closed ones while maintaining transverse properties

Basic Euler characteristic remains invariant under deformation

Positivity of Euler characteristic linked to large structural algebra

Abstract

We show that a compact manifold admitting a Killing foliation with positive transverse curvature fibers over finite quotients of spheres or weighted complex projective spaces, provided that the singular foliation defined by the closures of the leaves has maximal dimension. 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. Using this fact we prove that a Riemannian foliation of a compact manifold with finite fundamental group and nonvanishing Euler characteristic is closed. As another application we obtain that, for a positively curved Killing foliation of a compact manifold, if the structural algebra has sufficiently large dimension then the basic Euler characteristic is positive.