Representation of flat Lagrangian \(H\)-umbilical submanifolds in complex Euclidean spaces
In a previous paper [\textit{B. Y. Chen}, Isr. J. Math. 99, 69-108 (1997; Zbl 0884.53014)], the author introduced the notion of an \(H\)-umbilical Lagrangian submanifold of a complex space form. It is a Lagrangian submanifold for which the second fundamental form \(h\) can be written as \[ h(X,Y)= \alpha\langle JX, H\rangle\langle JY, H\rangle H+ \beta|H|^2\{\langle X, Y\rangle H+\langle JX, H\rangle JY+ \langle JY, H\rangle JX\} \] for some functions \(\alpha\), \(\beta\) on \(M\). Besides the totally geodesic Lagrangian submanifolds, \(H\)-umbilical Lagrangian submanifolds are the simplest possible Lagrangian submanifolds. In [\textit{B. Y. Chen}, Complex extensors and Lagrangian submanifolds in complex Euclidean spaces, Tôhoku Math. J. (2) 49, 277-297 (1997; Zbl 0877.53041)], the author obtained a classification of all non-flat Lagrangian \(H\)-umbilical submanifolds showing that it must either be a complex extensor or a pseudo-sphere. In the present paper, the author completes the classification by studying the flat \(H\)-umbilical Lagrangian submanifolds. He shows that they can all be constructed starting from a special Legendre curve in \(S^{2n-1}\) on an interval \(I\) together with one arbitrary function \(f: I\to I\). Note that a curve \(z\) is called a Legendre curve if there exist parallel normal vector fields \(P_3,\dots, P_n\) along the curve such that \[ z''(s)= i\lambda(s) z'(s)- z(s)- \sum^n_{j=3} a_j(s) P_j(s), \] for some real-valued functions \(\lambda, a_3,\dots, a_n\).
scientific article; zbMATH DE number 1321870