Spectral properties of four-dimensional compact flat manifolds

ROBERTO J. MIATELLO; RICARDO A. PODESTÁ

ANNALS OF GLOBAL ANALYSIS AND GEOMETRY

Springer

Año: 2006 vol. 29 p. 17 - 50

0232-704X

We study the spectral properties of a large class of compact flat Riemannian manifolds of dimension 4, namely, those  whose corresponding Bieberbach groups have the canonical lattice as translation lattice. By using the explicit expression of the heat trace of the Laplacian acting on p-forms, we determine all p-isospectral and L-isospectral pairs and we show that in this class ofmanifolds, isospectrality on functions and isospectrality on p-forms  for all values of p are equivalent to each other. The list shows  for any p, $0<p<4$, many p-isospectral pairs that are not isospectral on functions and have different lengths of closed geodesics. We also determine all length isospectral pairs (i.e. with the same length multiplicities), showing that there are two weak length isospectral pairs that are not length isospectral, and many pairs, p-isospectral for all p and not length isospectral.