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Spectral rigidity of a compact negatively curved manifold. (English) Zbl 0936.58013

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.

MSC:

58J53 Isospectrality
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C20 Global Riemannian geometry, including pinching
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