CR-submanifolds of chen-type two in non-flat complex space forms

Abstract

We prove some nonexistence results for certain families of CR-submanifolds of Chen-type two in complex space forms. For example, there exist no holomorphic submanifolds of the complex hyperbolic space which are of 2-type via the standard embedding by projectors. This is in contrast to the situation in complex projective space, which is known to contain Einstein-Kähler submanifolds of 2-type. We further show that there are no mass-symmetric ruled real hypersurfaces in ℂQm (4c) of Chen-type 2. Additionally, we characterize mass-symmetric totally real submanifolds of 2-type in terms of detailed intrinsic and extrinsic conditions and derive some corollaries for Lagrangian submanifolds.