  We revisit some uniqueness results for a geometric nonlinear PDE related to
the scalar curvature in Riemannian geometry and CR geometry. In the Riemannian
case we give a new proof of the uniqueness result assuming only a positive
lower bound for Ricci curvature. We apply the same principle in the CR case and
reconstruct the Jerison-Lee identity in a more general setting. As a
consequence we prove a stronger uniqueness result in the CR case. We also
discuss some open problems for further study.
