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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 Poincaré 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.

Riemannian nilmanifolds, the wave trace, and the length spectrum