On totally umbilical \(CR\)-submanifolds of a Kähler manifold (Q1210471)

Let \(M\) be a compact 3-dimensional totally umbilical \(CR\)-submanifold of a Kähler manifold \(\widetilde M\) with positive holomorphic sectional curvature. Then \(M\) has strictly positive sectional curvature when the mean curvature of \(M\) in \(\widetilde M\) is nowhere zero. Combining this with a classification theorem of Hamilton on compact 3-dimensional Riemannian manifolds with positive Ricci curvature, the author observes that if a compact 3-dimensional totally umbilical \(CR\)-submanifold \(M\) of a Kähler manifold with positive holomorphic sectional curvature has nowhere vanishing mean curvature, \(M\) is diffeomorphic to a 3-sphere, or to a real projective 3-space, or to a 3-dimensional lens space.