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This paper studies the CR-version of Obata theorem on a pseudo-Hermitian CR-manifold (M,θ). The main result of the paper is proving that CR―Obata theorem holds on real ellipsoid E(A) with contact form θ = 1/2i(∂ρ A ― ∂ρ A ), where ρ A (z) = |z| 2 + Re Σ n j=1 A j z 2 j ― 1 with A j ∈ (―1, 1).

On the CR–Obata theorem and some extremal problems associated to pseudoscalar curvature on the real ellipsoids in $\mathbb{C}^{n+1}$