The minimal marked length spectrum of Riemannian two-step nilmanifolds

Abstract

The purpose of this paper is to compare the minimal marked length spectrum and the Laplace spectrum on functions and on forms for Riemannian two-step nilmanifolds. A Riemannian nilmanifold is a closed manifold of the form ( \G, g), where G is a simply connected nilpotent Lie group, is a cocompact (i.e., \G compact) discrete subgroup of G, and g arises from a left invariant metric on G. Examples of nilmanifolds include flat tori and Heisenberg manifolds. The Laplace spectrum of a closed Riemannian manifold (M, g) is the set of eigenvalues of the Laplace–Beltrami operator , counted with multiplicity. The Laplace–Beltrami operator may be extended to act on smooth p-forms by = dδ + δd, where δ is the metric adjoint of d. Two manifolds have the same marked length spectrum if there exists an isomorphism between the fundamental groups such that corresponding free homotopy classes of loops can be represented by smoothly closed geodesics of the same length. Two manifolds have the same minimal marked length spectrum (resp., maximal marked length spectrum) if there exists an isomorphism between the fundamental groups such that the smallest (resp., longest) closed loops in corresponding free homotopy classes have the same length. The main theorem of this paper is the following (see Theorem 2.5).