Isospectrality in the FIO category (Q1203629)

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.