A brief note on the spectrum of the basic Dirac operator

TL;DR

This paper proves the invariance of the basic Dirac operator's spectrum under metric changes on Riemannian foliations, introduces new eigenvalue estimates, and defines a metric-independent basic Laplacian.

Contribution

It establishes spectral invariance of the basic Dirac operator, provides new eigenvalue bounds, and introduces a spectrum-independent basic Laplacian for Riemannian foliations.

Findings

Spectrum of the basic Dirac operator is invariant under metric changes.

New eigenvalue estimates relate to the O'Neill tensor and first eigenvalue.

A new basic Laplacian with metric-independent spectrum is defined.

Abstract

In this paper, we prove the invariance of the spectrum of the basic Dirac operator defined on a Riemannian foliation $(M,\mathcal{F})$ with respect to a change of bundle-like metric. We then establish new estimates for its eigenvalues on spin flows in terms of the O'Neill tensor and the first eigenvalue of the Dirac operator on $M$. We discuss examples and also define a new version of the basic Laplacian whose spectrum does not depend on the choice of bundle-like metric.