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, 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.

Riemannian flows and adiabatic limits

“…as f → 0 uniformly. Since the eigenvalues of D b,f are constant in f and are those of D b (see [14]), the eigenvalues of D M,f converge to those of D b , because the spectrum is continuous as a function of the operator norm (see Lemma 5.1 in the appendix).…”

Homotopy invariance of cohomology and signature of a Riemannian foliation

“…In [12], we showed the invariance of the spectrum of D b with respect to a change of metric on M in any way that leaves the transverse metric on the normal bundle intact (this includes modifying the subbundle Q ⊂ T M, as one must do in order to make the mean curvature basic, for example). That is, Theorem 2.1.…”

“…That is, Theorem 2.1. (In [12]) Let (M, F ) be a compact Riemannian manifold endowed with a Riemannian foliation and basic Clifford bundle E → M. The spectrum of the basic Dirac operator is the same for every possible choice of bundle-like metric that is associated to the transverse metric on the quotient bundle Q.…”

The index of a transverse Dirac-type operator: the case of abelian Molino sheaf

“…Given the transverse metric, it was shown in [25] that the spectrum is independent of the particular choice of bundle-like metric.…”

“…Still working on M, the problem of self-adjointness was resolved by means of a modified basic Dirac-type operator, involving the basic projection of κ [25]. Given the transverse metric, it was shown in [25] that the spectrum is independent of the particular choice of bundle-like metric.…”