Point Leaf Maximal Singular Riemannian Foliations in Positive Curvature

TL;DR

This paper extends the concept of fixed point homogeneous actions to singular Riemannian foliations, revealing that positively curved manifolds with these foliations are often topologically similar to symmetric spaces or their quotients.

Contribution

It introduces the notion of point leaf maximal SRFs in positive curvature and classifies the topology of manifolds admitting such foliations, including non-homogeneous examples.

Findings

Manifolds with point leaf maximal SRFs are often diffeomorphic or homeomorphic to compact rank one symmetric spaces.

Such manifolds are cohomology CROSSes or finite quotients of them.

Existence of non-homogeneous SRFs on non-simply connected manifolds.

Abstract

We generalize the notion of fixed point homogeneous isometric group actions to the context of singular Riemannian foliations. We find that in some cases, positively curved manifolds admitting these so-called point leaf maximal SRF's are diffeo/homeomorphic to compact rank one symmetric spaces. In all cases, manifolds admitting such foliations are cohomology CROSSes or finite quotients of them. Among non-simply connected manifolds, we find examples of such foliations which are non-homogeneous