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Spectral asymmetry and Riemannian geometry. III

Published online by Cambridge University Press:  24 October 2008

M. F. Atiyah ,
V. K. Patodi  and
I. M. Singer
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
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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
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 79 , Issue 1 , January 1976 , pp. 71 - 99
Copyright
Copyright © Cambridge Philosophical Society 1976

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References

REFERENCES

(1)Atiyah, M. F., K-Theory (Benjamin; New York, 1967).Google Scholar
(2)Atiyah, M. F., Bott, R. and Patodi, V. K.On the heat equation and the index theorem. Inventiones math. 19 (1973), 279–330.CrossRefGoogle Scholar
(3)Atiyah, M. F., Patodi, V. K. and Singer, I. M.Spectral asymmetry and Riemannian geometry. Bull. Loud. Math. Soc. 5 (1973), 229–34.CrossRefGoogle Scholar
(4)Atiyah, M. F., Patodi, V. K. and Singer, I. M.Spectral asymmetry and Riemannian geometry. I. Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69.CrossRefGoogle Scholar
(5)Atiyah, M. F., Patodi, V. K. and Singer, I. M.Spectral asymmetry and Riemannian geometry. II. Math. Proc. Cambridge Philos. Soc. 78 (1975), 405–432.CrossRefGoogle Scholar
(6)Atiyah, M. F. and Hirzebruch, F.Vector bundles and homogeneous spaces. Proc. Symposium in Pure Math. Vol. 3, Amer. Math. Soc. (1961).CrossRefGoogle Scholar
(7)Atiyah, M. F. and Singer, I. M.The index of elliptic operators. I. Ann. of Math. 87 (1968), 484–530.CrossRefGoogle Scholar
(8)Atiyah, M. F. and Singer, I. M.The index of elliptic operators. III. Ann. of Math. 87 (1968), 546–604.CrossRefGoogle Scholar
(9)Atiyah, M. F. and Singer, I. M.The index of elliptic operators. IV. Ann. of Math. 93 (1971), 119–38.CrossRefGoogle Scholar
(10)Atiyah, M. F. and Singer, I. M.The index of elliptic operators. V. Ann. of Math. 93 (1971), 139–49.CrossRefGoogle Scholar
(11)Atiyah, M. F. and Singer, I. M.Index theory for skew-adjoint Fredholm operators. Publ. Math. Inst. Hautes Etudes Sci. (Paris), No. 37 (1969).CrossRefGoogle Scholar
(12)Bott, R.The stable homotopy of the classical groups. Ann. of Math. 70 (1959), 313–37.CrossRefGoogle Scholar
(13)Chern, S. and Simons, J.Characteristic forms and geometric invariants. Ann. of Math. 99 (1974), 48–69.Google Scholar
(14)Seeley, R. T.Complex powers of an elliptic operator. Proc. Symposium in Pure Math. Vol. 10, Amer. Math. Soc. (1967), 288–307.CrossRefGoogle Scholar