Interaction of Legendre curves and Lagrangian submanifolds
Let \(\widetilde M^n(4c)\) be a complex space form of holomorphic sectional curvature \(4c\neq 0\) and \(\pi:S^{2n+1} (c)\to \widetilde M^n(4c)\) the Hopf fibration from the sphere of radius \(1/ \sqrt c\) for \(c>0\) resp. from the anti-de Sitter space of curvature \(c<0\). The totally real submanifolds \(M\subset \widetilde M^n(4c)\) of (real) dimension \(n\) are called Lagrangian submanifolds. Locally there exists a 1:1-correspondence to the \(n\)-dimensional horizontal submanifolds in \(S^{2n+1}(c)\) with respect to \(\pi\). For \(n=1\) the latter submanifolds are called Legendre curves. Besides the totally geodesic submanifolds, the simplest Lagrangian submanifolds are the \(H\)-umbilical ones, whose shape operator has a special, simple structure. In this article a classification of the latter class is obtained. It is shown that many of the examples can be constructed by starting from a Legendre curve, deriving a horizontal warped product in \(S^{2n+1} (c)\) from it and then projecting this submanifold into \(\widetilde M^n(4c)\).
scientific article; zbMATH DE number 1062226