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In this paper, we obtain a sharp lower bound estimate for diameters with respect to an adapted metric in closed pseudohermitian (2n + 1)manifolds when a sharp lower bound estimate for the first positive eigenvalue of the sublaplacian is achieved. As a consequence, we confirm the CR Obata Conjecture on a closed pseudohermitian (2n+ 1)-manifold with an extra condition on covariant derivatives of torsion.

The diameter estimate and its application to CR Obata’s Theorem on closed pseudohermitian $(2n+1)$-manifolds