Modified differentials and basic cohomology for Riemannian foliations

TL;DR

This paper introduces a new exterior derivative for basic forms in Riemannian foliations, leading to a basic cohomology satisfying Poincaré duality and revealing links between curvature, tautness, and characteristic vanishing.

Contribution

It proposes a novel twisted basic cohomology for Riemannian foliations that satisfies Poincaré duality in the transversally orientable case.

Findings

Basic Euler characteristic can vanish under certain curvature conditions.

Basic signature relates to curvature and tautness.

New cohomology framework enhances understanding of foliation geometry.

Abstract

We define a new version of the exterior derivative on the basic forms of a Riemannian foliation to obtain a new form of basic cohomology that satisfies Poincar\'e duality in the transversally orientable case. We use this twisted basic cohomology to show relationships between curvature, tautness, and vanishing of the basic Euler characteristic and basic signature.