Riemannian nilmanifolds, the wave trace, and the length spectrum

This paper examines the length spectrum on two-step nilmanifolds toward determining what, exactly, the wave trace says about isospectral manifolds. In particular, for each length occurring in the length spectrum of a two-step nilmanifold, we compute the leading order term in the associated wave invariant, under the assumption of the clean intersection hypothesis. En route, we calculate the Poincar´e or First Return map for all two-step nilmanifolds. As an application, we explain certain examples of Heisenberg manifolds constructed by C.S. Gordon (C.S. Gordon, The Laplace spectra versus the length spectra of Riemannian manifolds, Contemporary mathematics: nonlinear problems in geometry (Mobile, Ala., 1985) vol. 51, AMS, 1986, pp. 63-80.) that are isospectral on functions, but have different multiplicities in the length spectrum. The multiplicity of a length is defined here as the number of free homotopy classes of loops that can be represented by a closed geodesic of that length.

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Published 1 January 2008