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Spectral asymmetry and Riemannian geometry. III

Published online by Cambridge University Press:  24 October 2008

M. F. Atiyah
Affiliation:
Oxford University
V. K. Patodi
Affiliation:
Tata Institute for Fundamental Research, Bombay
I. M. Singer
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts

Extract

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 ηA(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 ηA(0), which is a measure of ‘spectral asymmetry’, was studied in detail particularly in relation to representations of the fundamental group.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1976

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References

(1)Atiyah, M. F., K-Theory (Benjamin; New York, 1967).Google Scholar
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