On Lagrangian \(H\)-umbilical surfaces in \(\mathbb{C} P^2(\widetilde {c})\) (Q5939536)

It is known that there exist no totally umbilical Lagrangian submanifolds in a complex space form \(M^{n}(4c)\), \(n\geq 2,\) other than the totally geodesic ones [\textit{B-Y. Chen} and \textit{K. Ogiue}, Mich. Math. J. 21, 225--229 (1974; Zbl 0295.53028)]. On the other hand, B-Y. Chen introduced the notion of Lagrangian \(H\)-umbilical submanifolds, which are the ``simplest'' submanifolds close to the totally geodesic ones in \(M^{n}(4c)\), and showed that except some exceptional cases, Lagrangian \(H\)-umbilical submanifolds of \(\mathbb{C} P^{n}\) can be obtained from Legendre curves in \(S^{3}\) via warped products [\textit{B-Y. Chen}, Isr. J. Math. 99, 69--108 (1997; Zbl 0884.53014)]. In this note, the author shows that a Lagrangian \(H\)-umbilical surface of \(\mathbb{C} P^{2}(c)\) is isotropic, if and only if, it is minimal. The proof is derived from the Gauss equation and a formula previously proved by the author concerning the length of the second fundamental form. By using this result, he obtains as a corollary that a Lagrangian \(H\)-umbilical surface of \(\mathbb{C} P^{2}(c)\) which, in addition, either is constant isotropic or has constant normal scalar curvature, must have constant Gauss curvature and, therefore, it must be locally congruent to a flat torus by a result of \textit{S. Maeda} [Isotropic immersions, Can. J. Math. 38, 416--430 (1986; Zbl 0576.53013)].