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We prove a CR Obata type result that if the first positive eigenvalue of thesub-Laplacian on a compact strictly pseudoconvex pseudohermitian manifold witha divergence free pseudohermitian torsion takes the smallest possible valuethen, up to a homothety of the pseudohermitian structure, the manifold is thestandart Sasakian unit sphere. We also give a version of this theorem using theexistence of a function with traceless horizontal Hessian on a complete, withrespect to Webster's metric, pseudohermitian manifold.

An Obata type result for the first eigenvalue of the sub-Laplacian on a CR manifold with a divergence free torsion