Spectral asymmetry and Riemannian geometry. III

Abstract

In Parts I and II of this paper ((4),(5)) we studied the 'spectral asymmetry' of certain elliptic self-adjoint operators arising in Riemannian geometry. More precisely, for any elliptic self-adjoint operator A on a compact manifold we defined η_A(s)=Σ_λ+0signλ|λ|^-8, where λ runs over the eigenvalues of A. For the particular operators of interest in Riemannian geometry we showed that η_A(s) had an analytic continuation to the whole complex s-plane, with simple poles, and that s=0 was not a pole. The real number η_A(0), which is a measure of 'spectral asymmetry', was studied in detail particularly in relation to representations of the fundamental group.