A brief note on the spectrum of the basic Dirac operator

Georges Habib (1) , Ken Richardson (2)

1 MPI-MiS - Max Planck Institute for Mathematics in the Sciences
2 Department of Mathematics and Statistics [Texas Tech]

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.

Domaines: Géométrie différentielle [math.DG]

Georges Habib, Ken Richardson. A brief note on the spectrum of the basic Dirac operator. 2008. ⟨hal-00321431⟩