Spectral triples and differential calculi related to the Kronecker foliation

Following the ideas of Connes and Moscovici, we describe two spectral triples related to the Kronecker foliation, whose generalized Dirac operators are related to first and second order signature operators. We also consider the corresponding differential calculi Omega(D), which are drastically different in the two cases. For the second order signature operator we calculate the Chem character of the spectral triple and the Dixmier trace of certain powers of its Dirac operator. As a side-remark, we give a description of a known calculus on the two-dimensional noncommutative torus in terms of generators and relations. (C) 2002 Published by Elsevier Science B.V.