Spectral theory of the Atiyah-Patodi-Singer operator on compact flat manifolds

ROBERTO J. MIATELLO; RICARDO A. PODESTÁ

THE JOURNAL OF GEOMETRIC ANALYSIS

SPRINGER

Año: 2011 p. 1 - 22

1050-6926

We study the spectral theory of the Dirac-type boundary operator D defined by Atiyah, Patodi and Singer, acting on smooth even forms of a compact at Riemannian manifold M. We give an explicit formula for the multiplicities of the eigenvalues of D in terms of values of characters of exterior representations of SO(n), where n = dimM. As a consequence, we give large families of D-isospectral at manifolds that are nonhomeomorphic to each other. Furthermore, we derive expressions for the eta series in terms of special values of Hurwitz zeta functions and, as a result, we obtain a simple explicit expression of the eta invariant.