Eta invariants of Dirac operators on locally symmetric manifolds
type: Item
instance of: scholarly article
title: Eta invariants of Dirac operators on locally symmetric manifolds
zbMATH Open document ID: 0672.58043
DOI: 10.1007/BF01393895
author: Henri Moscovici
author: Robert J. Stanton
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143671
review text: The eta-invariant of a self-adjoint elliptic differential operator on a compact manifold X was introduced by Atiyah, Patodi and Singer in connection with the index theorem for manifolds with boundary. It is a spectral invariant which measures the asymmetry of the spectrum of such an operator A. (Formally, \(\eta\) (A) is equal to the number of positive eigenvalues of A minus the number of negative eigenvalues). \(\eta\)- invariants of Dirac operators are closely related to several important invariants from differential topology. For X a compact oriented (4n-1)-dimensional Riemannian manifold of constant negative curvature, \textit{J. J. Millson} [Ann. Math., II. Ser. 108, 1-39 (1978; Zbl 0399.58020)] has proved a remarkable formula relating \(\eta_ X\) to the closed geodesics on X, where \(\eta_ X=\eta (B)\), B is tangential signature operator, acting on the even forms of X. The appropriate class of Riemannian manifolds for which a result of this type can be expected is that of non-positively curved locally symmetric manifolds, while the class of self adjoint operators whose \(\eta\)- invariants are interesting to compute is that of Dirac-type operators, eventually with additional coefficients in locally flat bundles. It is the purpose of this paper to formulate and prove such an extension of Millson's formula.
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 58J70
zbMATH DE Number: 4100247
zbMATH Keywords: eta-invariant
zbMATH Keywords: self-adjoint elliptic differential operator
zbMATH Keywords: spectral invariant
zbMATH Keywords: Dirac operators
MaRDI profile type: Publication
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OpenAlex ID: W2055665289
description: scientific article; zbMATH DE number 4100247