Spectral properties of flat manifolds

ROBERTO JORGE MIATELLO, JUAN PABLO ROSSETTI

CONTEMPORARY MATHEMATICS

American Mathematical Society

Lugar: Providence; Año: 2009 vol. 341 p. 83 - 115

0271-4132

Abstract. In this article we study spectral properties of  flat Riemannian manifolds, particularly, the determination of their p-form spectrum and the relations among different kinds of isospectralities: p-isospectrality, Sunada isospectrality and length isospectrality (i.e. with respect to lengths of closed geodesics). We present the methods, main results and examples obtained in several previous articles, and also some new material on self-intersections of closed geodesics in this setting. We give necessary and suffient conditions for a closed geodesic to self-intersect, some applications to the existence of simple and non-simple closed geodesics, and an example that shows that two`strongly´ isospectral manifolds can behave very differently in this respect.In this article we study spectral properties of  flat Riemannian manifolds, particularly, the determination of their p-form spectrum and the relations among different kinds of isospectralities: p-isospectrality, Sunada isospectrality and length isospectrality (i.e. with respect to lengths of closed geodesics). We present the methods, main results and examples obtained in several previous articles, and also some new material on self-intersections of closed geodesics in this setting. We give necessary and suffient conditions for a closed geodesic to self-intersect, some applications to the existence of simple and non-simple closed geodesics, and an example that shows that two`strongly´ isospectral manifolds can behave very differently in this respect.