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Spectral asymmetry and Riemannian geometry. III

  • M. F. Atiyah (a1), V. K. Patodi (a2) and I. M. Singer (a3)

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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.

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(1)Atiyah, M. F., K-Theory (Benjamin; New York, 1967).
(2)Atiyah, M. F., Bott, R. and Patodi, V. K.On the heat equation and the index theorem. Inventiones math. 19 (1973), 279330.
(3)Atiyah, M. F., Patodi, V. K. and Singer, I. M.Spectral asymmetry and Riemannian geometry. Bull. Loud. Math. Soc. 5 (1973), 229–34.
(4)Atiyah, M. F., Patodi, V. K. and Singer, I. M.Spectral asymmetry and Riemannian geometry. I. Math. Proc. Cambridge Philos. Soc. 77 (1975), 4369.
(5)Atiyah, M. F., Patodi, V. K. and Singer, I. M.Spectral asymmetry and Riemannian geometry. II. Math. Proc. Cambridge Philos. Soc. 78 (1975), 405432.
(6)Atiyah, M. F. and Hirzebruch, F.Vector bundles and homogeneous spaces. Proc. Symposium in Pure Math. Vol. 3, Amer. Math. Soc. (1961).
(7)Atiyah, M. F. and Singer, I. M.The index of elliptic operators. I. Ann. of Math. 87 (1968), 484530.
(8)Atiyah, M. F. and Singer, I. M.The index of elliptic operators. III. Ann. of Math. 87 (1968), 546604.
(9)Atiyah, M. F. and Singer, I. M.The index of elliptic operators. IV. Ann. of Math. 93 (1971), 119–38.
(10)Atiyah, M. F. and Singer, I. M.The index of elliptic operators. V. Ann. of Math. 93 (1971), 139–49.
(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).
(12)Bott, R.The stable homotopy of the classical groups. Ann. of Math. 70 (1959), 313–37.
(13)Chern, S. and Simons, J.Characteristic forms and geometric invariants. Ann. of Math. 99 (1974), 4869.
(14)Seeley, R. T.Complex powers of an elliptic operator. Proc. Symposium in Pure Math. Vol. 10, Amer. Math. Soc. (1967), 288307.

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