It is a well-known result that Gauss, Codazzi and Ricci equations are necessary and suficiently conditions to guarantee the existence of an isometric immersion of a given $n$-dimensional real Riemannian manifold into a $(n+p)$-dimensional real space form.

I would like to know if there is some version of this result for complex space forms. Can someone give me a reference?

givensare in the complex space form theorem. In the real case, the givens are a manifold $M^n$, a metric $g$ on $M$, an Euclidean vector bundle $N$ of rank $p$ over $M$ endowed with an Euclidean connection, and a section $I\!I$ of $N\otimes S^2(T^*M)$, all assumed to satisfy Gauss, Codazzi, and Ricci. In the complex case, one also needs an orthogonal complex structure $J$ on $TM\oplus N$ plus some equations that relate it to the other data. It is only at this point that Cartan's Fundamental Lemma takes over. $\endgroup$1more comment