Surfaces in Euclidean 3-space with Maslovian normal bundles

The normal bundle T-perpendicular to M of a surface M in R-3 can be naturally immersed in C-3 = R-3 x R-3 as a Lagrangian submanifold. Let H denote the mean curvature vector field of T-perpendicular to M in C-3, and let J denote the complex structure of C-3. Then T-perpendicular to M is called Maslovian if H is nowhere zero and J H is a principal direction of the shape operator with respect to H. In this paper, we prove that a surface in R-3 has Maslovian normal bundle if and only if it is a part of a round sphere, a circular cylinder, or a circular cone.