Comparisons of Laplace spectra, length spectra and geodesic flows of some Riemannian manifolds.(English)Zbl 0837.58035

The goal of this article is to discuss relationships between the Laplace spectrum, the length spectrum and the geodesic flow of some compact Riemannian manifolds, particularly, hyperbolic manifolds and nilmanifolds. The authors have explained why the resulting Laplace isospectral manifolds have the same weak length spectra and also have illustrated these ideas by proving a new result:
Theorem A. Compact orientable strongly isospectral hyperbolic manifolds always have the same length spectra.
Theorem B. There exists a class $$\mathcal M$$ of two-step nilmanifolds such that any nilmanifold whose geodesic flow is $$C^0$$-conjugate to an element $$M$$ of $$\mathcal M$$ must be isometric to $$M$$. Included in $$\mathcal M$$ are continuous families of strongly Laplace isospectral manifolds.