  We study the (massless) Dirac operator on a 3-sphere equipped with Riemannian
metric. For the standard metric the spectrum is known. In particular, the
eigenvalues closest to zero are the two double eigenvalues +3/2 and -3/2. Our
aim is to analyse the behaviour of eigenvalues when the metric is perturbed in
an arbitrary smooth fashion from the standard one. We derive explicit
perturbation formulae for the two eigenvalues closest to zero, taking account
of the second variations. Note that these eigenvalues remain double eigenvalues
under perturbations of the metric: they cannot split because of a particular
symmetry of the Dirac operator in dimension three (it commutes with the
antilinear operator of charge conjugation). Our perturbation formulae show that
in the first approximation our two eigenvalues maintain symmetry about zero and
are completely determined by the increment of Riemannian volume. Spectral
asymmetry is observed only in the second approximation of the perturbation
process. As an example we consider a special family of metrics, the so-called
generalized Berger spheres, for which the eigenvalues can be evaluated
explicitly.
