Eta invariants of Dirac operators on locally symmetric manifolds (Q1120176)

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.