Croke, Christopher B.; Sharafutdinov, Vladimir A. Spectral rigidity of a compact negatively curved manifold. (English) Zbl 0936.58013 Topology 37, No. 6, 1265-1273 (1998). A Riemannian manifold is called spectrally rigid if it does not admit a nontrivial isospectral deformation. An isospectral deformation is trivial if it is implemented by a one-parameter family of diffeomorphisms. The notion of spectral rigidity was introduced by V. Guillemin and D. Kazhdan [Topology 19, 301-312 (1980; Zbl 0465.58027) and Proc. Symp. Pure Math. 36, 153-180 (1980; Zbl 0456.58031)]. Guillemin and Kazhdan proved that compact negatively curved \(2\)-manifolds are spectrally rigid. They proved the same result for compact negatively curved \(n\)–manifolds under a pointwise curvature pinching assumption. The paper under review proves the result in general. The main results are: A compact negatively curved Riemannian manifold is spectrally rigid. Let \(M\) be a compact negatively curved Riemannian manifold with simple length spectrum. If \(\Delta+q_1\) and \(\Delta+q_2\) have the same spectrum for \(q_1,q_2\in C^\infty(M)\) then \(q_1=q_2\). Both results follow from a common source which is now proved for all compact negatively curved manifolds: a symmetric tensor field has a primitive if certain integrals over closed geodesics vanish. The fact that this implies the two results above was already observed by Guillemin and Kazhdan. Reviewer: Matthias Lesch (Tucson) Cited in 42 Documents MSC: 58J53 Isospectrality 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C20 Global Riemannian geometry, including pinching Keywords:isospectral deformation; spectral rigidity; compact negatively curved Riemannian manifold Citations:Zbl 0465.58027; Zbl 0456.58031 PDFBibTeX XMLCite \textit{C. B. Croke} and \textit{V. A. Sharafutdinov}, Topology 37, No. 6, 1265--1273 (1998; Zbl 0936.58013) Full Text: DOI