Length and eigenvalue equivalence

Let M be a compact Riemannian manifold, and let ∆ = ∆M denote the Laplace– Beltrami operator of M acting on L2(M). The eigenvalue spectrum E (M) consists of the eigenvalues of ∆ listed with their multiplicities. Two manifolds M1 and M2 are said to be isospectral if E (M1) = E (M2). Geometric and topological constraints are forced on isospectral manifolds; for example if the manifolds are hyperbolic (complete with all sectional curvature equal to −1) then they must have the same volume [18], and so for surfaces the same genus. Another invariant of M is the length spectrum L (M) of M; that is the set of all lengths of closed geodesics on M counted with multiplicities. Two manifolds M1 and M2 are said to be iso-length spectral if L (M1) = L (M2). Under the hypothesis of negative sectional curvature the invariants E (M) and L (M) are closely related. For example, it is known that E (M) determines the set of lengths of closed geodesics, and in the case of closed hyperbolic surfaces, the stronger statement that E (M) determines L (M) and vice-versa holds [7, 8]. In this paper we address the issue of how much information is lost by forgetting multiplicities. More precisely, for a compact Riemannian manifold M, define the eigenvalue set (resp. length set and primitive length set) to be the set of eigenvalues of ∆ (resp. set of lengths all closed geodesics and lengths of all primitive closed geodesics) counted without multiplicities. These sets will be denoted E(M), L(M) and Lp(M) respectively. Two manifolds M1 and M2 are said to be eigenvalue equivalent (resp. length equivalent and primitive length equivalent) if E(M1) = E(M2) (resp. L(M1) = L(M2) and Lp(M1) = Lp(M2)). Although length ∗Partially supported by the N. S. F. †Partially supported by a C.M.I. lift-off ‡Partially supported by the N. S. F. §Partially supported by the N. S. F.