Core reduction for singular Riemannian foliations and applications to positive curvature

We expand upon the notion of a pre-section for a singular Riemannian foliation (M, F) i.e. a proper submanifold N subset of M retaining all the transverse geometry of the foliation. This generalization of a polar foliation provides a similar reduction, allowing one to recognize certain geometric or topological properties of (M, F) and the leaf space M/F. In particular, we show that if a foliated manifold M has positive sectional curvature and contains a nontrivial pre-section, then the leaf space M/F has nonempty boundary. We recover as corollaries the known result for the special case of polar foliations as well as the well-known analogue for isometric group actions.