Spectral rigidity of a compact negatively curved manifold

In [1] V. Guillemin and D. Kazhdan introduced the following definition of spectral rigidity of a Riemannian manifold. Let (M, g) be a compact boundaryless Riemannian manifold. A family g of Riemannian metrics on M smoothly depending on the parameter τ ∈ [−2, 2] is called the deformation of the metric g if g = g. A deformation is called trivial if there exists a one-parameter family of diffeomorphisms φ : M → M such that φ = Id, and g = (φ )∗g0. Given a deformation g (−2 ≤ τ ≤ 2), let ∆ : C∞(M) → C∞(M) be the Laplace-Beltrami operator corresponding to the metric g . The deformation is called isospectral if, for all −2 ≤ τ ≤ 2 spectra of the operators ∆ and ∆ coincide (counting multiplicities). A Riemannian manifold (M, g) is called spectrally rigid if it does not admit non-trivial isospectral deformations. The main result of the present article is the following