A brief note on the spectrum of the basic Dirac operator

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.

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• Seiberg-Witten invariants on manifolds with Riemannian foliations of codimension 4
• The equivariant index theorem for transversally elliptic operators and the basic index theorem for Riemannian foliations
• Transverse Killing and twistor spinors associated to the basic Dirac operators
• Perturbations of basic Dirac operators on Riemannian foliations