The Index of Hypoelliptic Operators on Foliated Manifolds

TL;DR

This paper develops an index theorem for hypoelliptic operators on foliated manifolds, extending Connes' tangent groupoid approach used in the Atiyah-Singer index theorem, and provides a largely self-contained proof.

Contribution

It introduces a new index theorem for hypoelliptic operators on foliated manifolds using a tangent groupoid approach, expanding the scope of index theory.

Findings

Established an index formula for hypoelliptic operators on foliated manifolds

Extended Connes' tangent groupoid method to hypoelliptic settings

Provided a self-contained proof of the index theorem

Abstract

We present an index theorem for certain hypoelliptic differential operators on foliated manifolds. Our proof is a development of Alain Connes tangent groupoid proof of the Atiyah-Singer index theorem. The paper is largely self-contained.