## The Laplace spectra versus the length spectra of Riemannian manifolds.(English)Zbl 0591.53042

Nonlinear problems in geometry, Proc. AMS Spec. Sess., 820th Meet. AMS, Mobile/Ala. 1985, Contemp. Math. 51, 63-80 (1986).
[For the entire collection see Zbl 0579.00012.]
Let M be a compact Riemannian manifold and $$\Delta$$ the Laplace operator acting on smooth functions. This paper concerns the following problem of spectral geometry: can we say that the $$\Delta$$-spectrum of M determines the length spectrum of M, i.e. the collection of lengths of closed geodesics in M ? One can define the multiplicity of a length $$\lambda$$ as the number of closed geodesics of length $$\lambda$$, or as the number of free homotopy classes of loops for which the shortest loop has length $$\lambda$$. In section 1 the author considers examples of continuous families of $$\Delta$$-isospectral, non-isometric manifolds, and he shows that these manifolds are also length-isospectral with either definition. In section 2 he considers Riemannian Heisenberg manifolds and proves that for these manifolds $$\Delta$$-isospectrality implies the length- isospectrality under the first definition of length spectrum but frequently not under the second definition.
Reviewer: D.Perrone

### MSC:

 53C20 Global Riemannian geometry, including pinching 53C22 Geodesics in global differential geometry 58J50 Spectral problems; spectral geometry; scattering theory on manifolds

Zbl 0579.00012