Eigenvalues of the Laplace Operator on Certain Manifolds.

To every compact Riemannian manifold M there corresponds the sequence 0 = X1 < X2 < X3 .< ... of eigenvalues for the Laplace operator on M. It is not known just how much information about M can be extracted from this sequence.' This note will show that the sequence does not characterize M completely, by exhibiting two 16-dimensional toruses which are distinct as Riemannian manifolds but have the same sequence of eigenvalues. By a flat torus is meant a Riemannian quotient manifold of the form Rn/L, where L is a lattice (= discrete additive subgroup) of rank n. Let L* denote the dual lattice, consisting of all y e R' such that x y is anl integer for all x e L. Then each y e L* determines an eigenfunction f(x) = exp(27r ix-y) for the Laplace operator on Rn/L. The corresponding eigenvalue X is equal to (27r)2y y. Hence, the number of eigenvalues less than or equal to (27rr)2 is equal to the number of points of L* lying within a ball of radius r about the origin. According to Witt2 there exist two self-dual lattices L1, L2 C R16 which are distinct, in the sense that no rotation of R16 carries L1 to L2, such that each ball about the origin contains exactly as many points of L1 as of L2. It follows that the Riemannian manifolds R16/L1 and R16/L2 are not isometric, but do have the same sequence of eigenvalues. In an attempt to distinguish R16/L1 from R16/L2 one might consider the eigenvalues of the Hodge-Laplace operator A = da + 6d, applied to the space of differential p-forms. However, both manifolds are flat and parallelizable, so the identity