On the Dirac spectrum of homogeneous 3-spheres

TL;DR

This paper proves that on 3-spheres, any two left-invariant metrics with identical Dirac spectra are actually the same up to isometry, and it computes eigenvalues for specific cases.

Contribution

It establishes spectral rigidity for left-invariant metrics on 3-spheres and computes eigenvalues for metrics with positive scalar curvature.

Findings

Spectral uniqueness of left-invariant metrics on S^3 and SO(3)

Explicit eigenvalue computations for positive scalar curvature cases

Extension of results to different spin structures

Abstract

We show that any two left-invariant metrics on $S^3\cong\operatorname{SU}(2)$ 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 $\operatorname{SO}(3)=S^3/\{\pm1\}$ for each of its two spin structures.