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

MIATELLO, ROBERTO J.; PODESTÁ RICARDO A.

THE JOURNAL OF GEOMETRIC ANALYSIS

SPRINGER

Año: 2012 vol. 22 p. 1027 - 1054

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 flat 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=dim M.As a consequence, we give large families of  D-isospectral flat 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.