# Ricci tensor in complex space forms

Let \begin{align} f:M^{2n}\to CQ_{c}^{N} \end{align} be an isometric immersion of　a Kähler manifold into a complex space form. We consider an orthonormal basis $Y=X_{1},..,X_{2n}$ and then calculate Ricci tensor, \begin{align} \operatorname{Ric}(Y)=\sum_{j=1}^{2n}<R'(X_{j},Y)Y,X_{j}>=\frac{c}{4}[2n-1+3|SY|^{2}]. \end{align} My question is how can I understand map $S$, the explanation in the paper by Dajczer and Rodriguez, On isometric immersions into complex space forms, is that $S:T_{x}M\to T_{x}M$ is given by $S=\pi\circ J'$, where $\pi : T_{x}CQ_{c}^{N}\to T_{x}M$ is the orthogonal projection. Is $S$ the tangent part of complex structure $J'$ on $CQ_{c}^{N}$?

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