The spectrum of basic Dirac operators

TL;DR

This survey explores basic Dirac operators on Riemannian foliations, highlighting their spectral properties and implications for transverse geometry, including a new adapted cohomology theory.

Contribution

It reviews the construction and spectral invariance of basic Dirac operators, introducing a transverse de Rham cohomology adapted to Riemannian foliations.

Findings

Spectral independence from bundle-like metric choices

Definition of a new transverse de Rham cohomology

Implications for transverse geometric analysis

Abstract

This is a survey article on a known generalization of Dirac-type operators to transverse operators called basic Dirac operators on Riemannian foliations, which are smooth foliations that have a transverse geometric structure. Construction of these operators requires the additional structure of what is called a bundle-like metric. We explain the result by Habib-R. that the spectrum of such an operator is independent of the choice of bundle-like metric, provided that the transverse geometric structure is fixed. We discuss consequences, which include defining a new version of the exterior derivative and de Rham cohomology that are nicely adapted to this transverse geometric setting.