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We study local in time Strichartz estimates for the Schrödinger equation associated to long range perturbations of the flat Laplacian on the Euclidean space. We prove that in such a geometric situation, outside a large ball centered at the origin, the solutions of the Schrödinger equation enjoy the same Strichartz estimates as in the nonperturbed situation. The proof is based on the Isozaki-Kitada parametrix construction. If in addition the metric is nontrapping, we prove that the Strichartz estimates hold in the whole space.