Spectral properties of 4dimensional compact flat manifolds
Abstract
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 of manifolds, isospectrality on functions and isospectrality on $p$forms for all values of $p$ are equivalent to each other. The list shows for any $p$, $1 \le p \le 3$, 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2005
 DOI:
 10.48550/arXiv.math/0505486
 arXiv:
 arXiv:math/0505486
 Bibcode:
 2005math......5486M
 Keywords:

 Differential Geometry;
 58J53;
 58C22;
 20H15
 EPrint:
 25 pages, several tables, to appear in AGAG