Quantum Ergodic Restriction Theorems: Manifolds Without Boundary

We prove that if ( M , g ) is a compact Riemannian manifold with ergodic geodesic flow, and if is a smooth hypersurface satisfying a generic microlocal asymmetry condition, then restrictions of an orthonormal basis of Δ-eigenfunctions of ( M , g ) to H are quantum ergodic on H . The condition on H is satisfied by geodesic circles, closed horocycles and generic closed geodesics on a hyperbolic surface. A key step in the proof is that matrix elements of Fourier integral operators F whose canonical relation almost nowhere commutes with the geodesic flow must tend to zero.