Applications of Connes' geodesic flow to trace formulas in noncommutative geometry

The "trace formula" of Chazarain, Duistermaat, and Guillemin expresses that the singularities of the distribution trace of the wave group on a compact Riemannian manifold X is included in the set of periods of the geodesic fl ow restricted to S*X. Most of the objects involved in this trace formula ha ve analogous in Connes' Non commutative Geometry. This paper shows, on seve ral significant examples of Noncommutative Geometry, that Connes' definitio n of geodesic flow leads to statements analogous to the classical trace for mula of Chazarain, Duistermaat, and Guillemin. (C) 1998 Academic Press.