Transversal Dirac operators on distributions, foliations, and G-manifolds: Lecture notes

Abstract

In these survey lectures, we investigate the geometric and analytic properties of transverse Dirac operators. In particular, we define a transverse Dirac operator associated to a distribution that is essentially self-adjoint (Prokhorenkov-R result). We describe the Habib-R Theorem showing that the invariance of the spectrum of a basic Dirac operator on a Riemannian foliation. The Bruening-Kamber-R theorems give Atiyah-Singer type formulas for the equivariant index of transversally elliptic operators on G-manifolds and the index of basic Dirac operators on Riemannian foliations. These notes contain exercises at the end of each subsection and are meant to be accessible to graduate students.

38 pages, 86 exercises, lectures given at CRM (Barcelona)

References in corpus (5)

• Modified differentials and basic cohomology for Riemannian foliations
• The eta invariant and equivariant index of transversally elliptic operators
• Desingularizing compact Lie group actions
• Transverse conformal Killing forms and a Gallot-Meyer Theorem for foliations
• Equivariant cohomology and resolution

Cited by in corpus (1)

• A Local Index Theorem of Transversal Type on Manifolds with Locally Free ${\mathbb{S}}^1$-action