Lagrangian \(H\)-umbilical submanifolds in quaternion Euclidean spaces (Q819275)

\textit{B. Y. Chen} and \textit{K. Ogiue} [Mich. Math. J. 21, 225--229 (1974; Zbl 0295.53028)] proved that there are no totally umbilical Lagrangian submanifolds in complex space forms, other than the totally geodesic ones. In this context, \textit{B. Y. Chen} introduced in [Isr. J. Math. 99, 69--108 (1997; Zbl 0884.53014)] the notion of Lagrangian \(H\)-umbilical submanifolds as the family of submanifolds closest in simplicity to the totally geodesic ones. Moreover, he has proven [Tohoku Math. J., II. Ser. 49, No. 2, 277--297 (1997; Zbl 0877.53041)] that the non-flat Lagrangian \(H\)-umbilical submanifolds in complex Euclidean space \( \mathbb{C} ^{n}, n\geq 3\), are either a Lagrangian pseudo-sphere, or a complex extensor of the unit hypersphere of \(\mathbb{E}^{n}\). In this paper, the authors consider the same type of questions for the case of the quaternionic Euclidean space \(\mathbb{H}^{n}\), \(n\geq 3\), obtaining results similar to those known in \(\mathbb{C} ^{n}\). The main result is that a non-flat Lagrangian \(H\)-umbilical isometric immersion in \(\mathbb{H}^{n}\), \(n\geq 3\), is, up to rigid motions of \(\mathbb{H}^{n}\), either a Lagrangian pseudo-sphere in \(\mathbb{C} ^{n}\), or a quaternion extensor of the unit hypersphere of \(\mathbb{E}^{n}\). They also get the corresponding version for \(n=2\) by imposing an extra condition.