On Lagrangian surfaces in CP2(c̃)

Abstract

Chen and Ogiue completely classified totally umbilical submanifolds in a non-flat complex-space-form. However, the classification problem of pseudo-umbilical submanifolds in a non-flat complex-space-form is still open. Very recently, Chen introduced the notion of Lagrangian H-umbilical submanifolds which is the simplest totally real submanifolds next to the totally geodesic ones in a complex-space-form and and classified Lagrangian H-umbilical submanifolds in a complex-space-form. The author proved that a Lagrangian H-umbilical surface M in a complex 2-dimensional complex projective space CP 2 (c) is an isotropic surface in CP 2 (c) if and only if M is a minimal surface in CP 2 (c). In this paper, firstly, we prove that a Lagrangian surface M in CP 2 (c) is an isotropic surface in CP 2 (c) if and only if M is a minimal surface in CP 2 (c). Secondly, we classify Lagrangian non-totally geodesic pseudo-umbilical surfaces in CP 2 (c).