## Isospectrality in the FIO category.(English)Zbl 0769.53026

Let $$(M_ i,g_ i)$$ be compact Riemannian manifolds and let $$\Delta_ i$$ be their Laplacians. They are said to be isospectral if $$\text{spec}(\Delta_ 1) = \text{spec}(\Delta_ 2)$$ where each eigenvalue is counted with multiplicity. This means that there is a unitary operator $$U: L_ 2(M_ 1) \to L_ 2(M_ 2)$$ which intertwines $$\Delta_ 1$$ and $$\Delta_ 2$$. If the intertwining operator is restricted to be a Fourier integral operator, the author shows the following geometric consequences hold: (a) If $$(M_ 1,g_ 1)$$ is nonpositively curved, then the $$(M_ i,g_ i)$$ have a common finite Riemannian cover. (b) If $$(M_ 1,g_ 1)$$ is a negatively curved surface with simple length spectrum, then the $$(M_ i,g_ i)$$ are isometric.
Reviewer: P.Gilkey (Eugene)

### MSC:

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