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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