On real local isometric immersions of {mathbb{C}Q^2_c} into {mathbb{C} P^{3}}

We investigate real local isometric immersions of Kähler manifolds ${mathbb{C}Q^2_c}$ of constant holomorphic curvature 4c into complex projective 3-space. Our main result is that the standard embedding of ${mathbb{C}P^2}$ into ${mathbb{C}P^3}$ has strong rigidity under the class of local isometric transformations. We also prove that there are no local isometric immersions of ${mathbb{C}Q^2_c}$ into ${mathbb{C}P^3}$ when they have different holomorphic curvature. An important method used is a study of the relationship between the complex structure of any locally isometric immersed ${mathbb{C}Q^2_c}$ and the complex structure of the ambient space ${mathbb{C}P^3}$ .