Dirac operators on foliations: the Lichnerowicz inequality

TL;DR

This paper develops Dirac operators on foliations using the Bismut-Lebeau localization technique, leading to improved lower bounds on the Laplacian and extending the Lichnerowicz-Hitchin vanishing theorem to foliated manifolds.

Contribution

It introduces a novel construction of Dirac operators on foliations with enhanced spectral properties, extending classical vanishing theorems to new geometric contexts.

Findings

Improved lower bounds for the Laplacian of Dirac operators on foliations.

Extension of the Lichnerowicz-Hitchin vanishing theorem to foliated manifolds.

Application of Bismut-Lebeau localization to foliation geometry.

Abstract

We construct Dirac operators on foliations by applying the Bismut-Lebeau analytic localization technique to the Connes fibration over a foliation. The Laplacian of the resulting Dirac operators has better lower bound than that obtained by using the usual adiabatic limit arguments on the original foliation. As a consequence, we prove an extension of the Lichnerowicz-Hitchin vanishing theorem to the case of foliations.