1995/10/05 by Ruth Gornet, Gornet, Ruth
Computer Science · Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Topological and Geometric Data Analysis #advanced mathematical theories #dg-ga #math.DG
paper · pdf · doi:10.48550/arxiv.dg-ga/9510001
amstex, 27 pages, no figures
arxiv created 1995/10/05 · openalex publication_date 1995/10/05 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the same marked length spectrum, they necessarily share the same Laplace spectrum on functions. In contrast, we present the first example of a pair of isospectral Riemannian manifolds with the same marked length spectrum but not the same spectrum on one-forms. Outside of the standard spheres vs. the Zoll spheres, which are not even isospectral, this is the only example of a pair of Riemannian manifolds with the same marked length spectrum, but not the same spectrum on forms. This partially extends and partially contrasts the work of Eberlein, who showed that on two-step nilmanifolds, the same marked length spectrum implies the same Laplace spectrum both on functions and on forms.