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The goal of this paper is to study periodic geodesics for sub-Riemannianmetrics on a contact 3D-manifold.We develop two rather independent subjects:1)The existence of closed geodesics spiraling around periodic Reeb orbits for ageneric metric.2) The precise study of the periodic geodesics for a rightinvariant metric on a quotient of SL2(R)

Periodic geodesics for contact sub-Riemannian 3D manifolds