On the Dirac Spectrum of Homogeneous 3-Spheres

Abstract

We show that any two left-invariant metrics on S^3cong {{text {SU}}}(2) which are isospectral for the associated classical Dirac operator D must be isometric. In the case of left-invariant metrics of positive scalar curvature, we compute and use the smallest eigenvalue of D^2. We show analogous results for left-invariant metrics on {{text {SO}}}(3)=S^3/{pm 1} for each of its two spin structures.