Abstract: In this paper, the authors prove that two types of totally real pseudo-umbilical submanifolds with flat normal bundle must be minimal and determine their shapes under the circumstances that the submanifolds are compact. Moreover, the authors also show that the normal bundle of a totally real and totally umbilical submanifold can't be flat.

Key words: Complex projective space, Totally real submanifolds, Pseudo-umbilical submanifolds, Flat normal bundle