DOI: 10.1017/s0305004100052105. Spectral asymmetry and Riemannian geometry. III

Spectral asymmetry and Riemannian geometry. III

10.1017/s0305004100052105

Crossref journal-article

Cambridge University Press (CUP)

Mathematical Proceedings of the Cambridge Philosophical Society (56)

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 definedwhere 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.

Bibliography

Atiyah, M. F., Patodi, V. K., & Singer, I. M. (1976). Spectral asymmetry and Riemannian geometry. III. Mathematical Proceedings of the Cambridge Philosophical Society, 79(1), 71â99.

Authors 3

M. F. Atiyah (first)
V. K. Patodi (additional)
I. M. Singer (additional)

References 14 Referenced 496

10.1090/pspum/003/0139181
10.2307/1970106
10.1007/BF01425417
10.1007/BF02684885
10.1112/blms/5.2.229
10.1017/S0305004100051872
10.2307/1971013
10.2307/1970756
{'key': 'S0305004100052105_ref001', 'volume-title': 'K-Theory', 'author': 'Atiyah', 'year': '1967'} / K-Theory by Atiyah (1967)
10.1017/S0305004100049410
10.2307/1970717
10.2307/1970757
10.1090/pspum/010/0237943
10.2307/1970715

Dates

Type	When
Created	16 years, 9 months ago (Nov. 7, 2008, 10:21 a.m.)
Deposited	6 years, 2 months ago (May 28, 2019, 5:22 p.m.)
Indexed	4 days, 10 hours ago (Aug. 21, 2025, 2:24 p.m.)
Issued	49 years, 7 months ago (Jan. 1, 1976)
Published	49 years, 7 months ago (Jan. 1, 1976)
Published Online	16 years, 10 months ago (Oct. 24, 2008)
Published Print	49 years, 7 months ago (Jan. 1, 1976)

Funders 0

None

BibTeX

@article{Atiyah_1976, title={Spectral asymmetry and Riemannian geometry. III}, volume={79}, ISSN={1469-8064}, url={http://dx.doi.org/10.1017/s0305004100052105}, DOI={10.1017/s0305004100052105}, number={1}, journal={Mathematical Proceedings of the Cambridge Philosophical Society}, publisher={Cambridge University Press (CUP)}, author={Atiyah, M. F. and Patodi, V. K. and Singer, I. M.}, year={1976}, month=jan, pages={71–99} }

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  "abstract": "<jats:p>In Parts I and II of this paper ((4), (5)) we studied the \u2018spectral asymmetry\u2019 of certain elliptic self-adjoint operators arising in Riemannian geometry. More precisely, for any elliptic self-adjoint operator <jats:italic>A</jats:italic> on a compact manifold we defined</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0305004100052105_eqn1\" /></jats:disp-formula></jats:p><jats:p>where \u03bb runs over the eigenvalues of <jats:italic>A</jats:italic>. For the particular operators of interest in Riemannian geometry we showed that \u03b7<jats:italic><jats:sub>A</jats:sub></jats:italic>(<jats:italic>s</jats:italic>) had an analytic continuation to the whole complex <jats:italic>s</jats:italic>-plane, with simple poles, and that <jats:italic>s</jats:italic> = 0 was not a pole. The real number \u03b7<jats:italic><jats:sub>A</jats:sub></jats:italic>(0), which is a measure of \u2018spectral asymmetry\u2019, was studied in detail particularly in relation to representations of the fundamental group.</jats:p>",
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