A normal line congruence and minimal ruled Lagrangian submanifolds in CPn

Abstract

We characterize Lagrangian submanifolds in complex projective space for which each parallel submanifold along normal geodesics with respect to a unit normal vector field is Lagrangian, by using a normal line congruence of the Lagrangian submanifold to complex 2-plane Grassmannian and quaternionic Kähler structure. As a special case, we can construct minimal ruled Lagrangian submanifolds in complex projective space from an austere hypersurface in sphere with non-vanishing Gauss-Kronecker curvature.