Singular Riemannian flows and characteristic numbers

TL;DR

This paper investigates singular Riemannian flows on even-dimensional closed manifolds, showing local foliation-diffeomorphism to isometric flows near singularities and deriving a residue formula for characteristic numbers.

Contribution

It introduces a method to compute characteristic numbers of manifolds via residues linked to singular Riemannian flows and establishes local equivalences near singular strata.

Findings

Local foliation-diffeomorphism to isometric flows near singularities

Residue formula for characteristic numbers

Application to singular Riemannian flows on closed manifolds

Abstract

Let $M$ be an even-dimensional, oriented closed manifold. We show that the restriction of a singular Riemannian flow on $M$ to a small tubular neighborhood of each connected component of its singular stratum is foliated-diffeomorphic to an isometric flow on the same neighborhood. We then prove a formula that computes characteristic numbers of $M$ as the sum of residues associated to the infinitesimal foliation at the components of the singular stratum of the flow.