Celebratio Mathematica — Singer — Complete Bibliography

Complete Bibliography

Works connected to Vijay Kumar Patodi

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M. F. Atiyah, V. K. Patodi, and I. M. Singer: “Spectral asymmetry and Riemannian geometry,” Bull. London Math. Soc. 5 (July 1973), pp. 229–234. MR 331443 Zbl 0268.58010 article

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M. F. Atiyah, V. K. Patodi, and I. M. Singer: “Spectral asymmetry and Riemannian geometry, I,” Math. Proc. Cambridge Philos. Soc. 77 (1975), pp. 43–69. MR 397797 Zbl 0297.58008 article

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M. F. Atiyah, V. K. Patodi, and I. M. Singer: “Spectral asymmetry and Riemannian geometry, II,” Math. Proc. Cambridge Philos. Soc. 78 : 3 (November 1975), pp. 405–432. MR 397798 Zbl 0314.58016 article

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M. F. Atiyah, V. K. Patodi, and I. M. Singer: “Spectral asymmetry and Riemannian geometry, III,” Math. Proc. Cambridge Philos. Soc. 79 : 1 (1976), pp. 71–99. MR 397799 Zbl 0325.58015 article

In Parts I and II of this paper [1975a; 1975b] 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 \[ \eta_A(s) = \sum_{\lambda\neq 0}\operatorname{sign} \lambda \,|\lambda|^{-s}, \] where \( \lambda \) runs over the eigenvalues of \( A \). For the particular operators of interest in Riemannian geometry we showed that \( \eta_A(s) \) had an analytic continuation to the whole complex \( s \)-plane, with simple poles, and that \( s = 0 \) was not a pole. The real number \( \eta_A(0) \), which is a measure of ‘spectral asymmetry’, was studied in detail particularly in relation to representations of the fundamental group.

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