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Small Data Scattering for the Nonlinear Schrödinger Equation on Product Spaces

We consider the cubic nonlinear Schrödinger equation, posed on ℝ n × M, where M is a compact Riemannian manifold and n ≥ 2. We prove that under a suitable smallness in Sobolev spaces condition on the data there exists a unique global solution which scatters to a free solution for large times.