each term on the right of (\[thm Ricci equ2\]). It turns out $$\begin{aligned}
        \nb_k(D_i\P_j)=&\nb_k(\nb_i\P_j)+\a u_{ik}\P_j+\a u_{jk}\P_i+\g \d_{ij}\nb_k\nb u,\nonumber\\
        \a u_k (D_i\P_j)=&\a^2u_ku_i\P_j+\a^2 u_ku_j\P_i+\a\g \d_{ij}u_k\nb u,\nonumber\\
        \a du(D_i\P_j)\P_k=& 2\a^2u_iu_j\P_k+\a\g \d_{ij}|\nb u|^2\P_k,\nonumber\\
        \g \bar g(\P_k,D_i\P_j)\nb u=&(\a\g u_i \d_{jk}+\a\g u_j\d_{ik})\nb u +\g^2\d_{ij}u_k\nb u.
        \label{thm Ricci equ3}
    \end{aligned}$$ Adding up these four terms into , we have $$\begin{aligned}
      D_k(D_i\P_j)=&\nb_k(\nb_i\P_j)+\a u_{ik}\P_j+\a u_{jk}\P_i+\g \d_{ij}\nb_k\nb u\nonumber\\
&+\a^2u_ku_i\P_j+\a^2 u_ku_j\P_i+\a\g \d_{ij}u_k\nb u\nonumber\\
 &+ 2\a^2u_iu_j\P_k+\a\g \d_{ij}|\nb u|^2\P_k\nonumber\\
  &+(\a\g u_i \d_{jk}+\a\g u_j\d_{ik})\nb u +\g^2\d_{ij}u_k\nb u.
\label{thm Ricci equ4}\end{aligned}$$ Using the metric tensor $\bar g$, we have $$\begin{aligned}
    \left< D_k(D_i\P_j),\P_k\right>_{\bar g}=&\left< \nb_k(\nb_i\P_j),\P_k\right>_{\bar g}+2\a u_{ij}+\g \bar\De u \d_{ij}\nonumber\\
    &+\big[\a^2+(2n+1)\a^2+2\a\g\big]u_iu_j\nonumber\\
    &+\big[(n+1)\a\g+\g^2\big]|\nb u|^2 \d_{ij}.
    \label{thm Ricci equ5}\end{aligned}$$ By swapping $k$ and $i$ in (\[thm Ricci equ4\]), we obtain $$\begin{aligned}
D_i(D_k\P_j)=&\nb_i(\nb_k\P_j)+\a u_{ik}\P_j+\a u_{ji}\P_k+\g \d_{kj}\nb_i\nb u\nonumber\\
&+\a^2u_ku_i\P_j+\a^2 u_iu_j\P_k+\a\g \d_{kj}u_i\nb u\nonumber\\
 &+ 2\a^2u_ku_j\P_i+\a\g \d_{kj}|\nb u|^2\P_i\nonumber\\
  &+(\a\g u_k \d_{ij}+\a\g u_j\d_{ik})\nb u +\g^2\d_{kj}u_i\nb u.
  \label{thm Ricci equ4.1}\end{aligned}$$ Similarly, we have $$\begin{aligned}
    \left< D_i(D_k\P_j),\P_k\right>_{\bar g} =&     \left< \nb_i(\nb_k\P_j),\P_k\right>_{\bar g}+\big[(n+1)\a+\g\big]u_{ij}+2\a\g|\nb u|^2\d_{ij}\nonumber\\ 
   &+\big[(n+3)\a^2+2\a\g+\g^2\big]u_iu_j.
  \label{thm Ricci equ5.1}\end{aligned}$$ Combining (\[thm Ricci equ5\]) and (\[thm Ricci equ5.1\]), using Definition \[defi Ricci\] we have $$\begin{aligned}
  R^D_{ij}= &\left< R^D(\P_k,\P_i)\P_j,\P_k\right>_{\bar g} \nonumber\\ 
    =&    \left< D_k(D_i\P_j),\P_k\right>_{\bar g}   - \left< D_i(D_k\P_j),\P_k\right>_{\bar g} \nonumber\\
    =&R_{ij}-\big[\a(n-1)+\g\big]u_{ij}+\big[(n-1)\a^2-\g^2\big]u_iu_j\nonumber\\
&+\g \bar\De u\d_{ij}+\big[(n-1)\a\g+\g^2\big]|\nb u|^2\d_{ij}.
    \label{thm Ricci equ 6}\end{aligned}$$ This finishes the proof for $Ric^D$.\
Part of the work 