Length equivalent hyperbolic manifolds

In the case when M is an orientable Riemannian manifold of negative curvature L (M) and L(M) are related to the eigenvalue spectrum of M; that is, the set E (M) of all eigenvalues of the Laplace–Beltrami operator acting on L2(M) counted with multiplicities. For example, in this setting it is known that E (M) determines L(M) (see [3]). In the case of closed hyperbolic surfaces, the stronger statement that E (M) determines L (M) and vice-versa holds [5, 6]. A pair of manifolds M1,M2 are said to be isospectral (resp. iso-length spectral) if E (M1) = E (M2) (resp. L (M1)L (M2)). Geometric and topological constraints are forced on isospectral manifolds, for example the manifolds have the same volume [10], and so for surfaces the same genus.