The Spectrum of Basic Dirac Operators

In this note, we discuss Riemannian foliations, which are smooth foliations that have a transverse geometric structure. We explain a known generalization of Dirac-type operators to transverse operators called “basic Dirac operators” on Riemannian foliations, which require the additional structure of what is called a bundle-like metric. We explain the result in [10] 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.