On pseudo-umbilical surfaces with nonzero parallel mean curvature vector in \(\mathbb{C} P^m (\widetilde{c})\) (Q2719783)

Let \(\mathbb{C}\mathbb{P}^m(\widetilde c)\) be a complex \(m\)-dimensional complex projective space with the Fubini-Study metric of constant holomorphic curvature \(\widetilde c\). The author proves the following main theorem: NEWLINENEWLINENEWLINELet \(M\) be a complete pseudo-umbilical surface with nonzero parallel mean curvature vector in \(\mathbb{C}\mathbb{P}^m (\widetilde c)\). If \(M\) is isotropic and the second fundamental form of \(M\) is commutative, then \(M\) is an extrinsic hypersphere in a 3-dimensional real projective space \(\mathbb{R} \mathbb{P}^3 (\widetilde c/4)\) of \(\mathbb{C}\mathbb{P}^3 (\widetilde c)\).