Spectral asymmetry and Riemannian geometry. III

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 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.

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