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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.
The primary emphasis of this article is on nilmanifolds and discussing their length spectra the authors have announced the following results:
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.
The proof of Theorem B is not included in this paper, since it is fairly technical. Instead of that a sketch in the simplest case is given in order to illustrate the geometric ideas.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C22 Geodesics in global differential geometry
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
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