Length minimizing geodesics and the length spectrum of Riemannian two-step nilmanifolds
The interest of the authors is focused on two-step nilpotent manifolds as closest to the Euclidean one and having clear algebraic structure quite determined by a \(J\)-operator. Due to the Gordon and Wilson technique of constructing isospectral manifolds of this type, which are not locally isometric, we have the possibility to examine the following key question: to what extent does the spectrum of the Laplacian determine the geometry? A major open question is whether there can exist examples of isospectral manifolds with different periods in the length spectrum. The authors extend their earlier result on the singular \(J\)-operator case and show against all hope that in the case the isospectrality necessarily yields the length spectrum coincidence. Relying on the work of \textit{C. Jang} and \textit{K. Park} [Geom. Dedicata 79, 65--80 (2000; Zbl 0943.22005)] they succeeded in calculating a lower bound on the injectivity radius of a general simply connected manifold of the above type. Investigating conjugate and cut points the authors find that they do not coincide for most geodesics in contrast to the one-dimensional center and Heisenberg type cases.
scientific article; zbMATH DE number 2013179
scientific article