A horizontal Chern-Gauss-Bonnet formula on totally geodesic foliations

TL;DR

This paper derives a new formula for the Euler characteristic of foliated Riemannian manifolds, expressing it solely in terms of transverse curvature invariants, using hypoelliptic sub-Laplacian techniques.

Contribution

It introduces a horizontal Chern-Gauss-Bonnet formula for totally geodesic foliations, linking topology to transverse curvature invariants.

Findings

Euler characteristic computed from transverse curvature invariants

Utilizes hypoelliptic sub-Laplacian on forms

Applicable under suitable geometric conditions

Abstract

Under suitable conditions, we show that the Euler characteristic of a foliated Riemannian manifold can be computed only from curvature invariants which are transverse to the leaves. Our proof uses the hypoelliptic sub-Laplacian on forms recently introduced by two of the authors.