was proved by Lichnerowicz [@Lic] using the classical Bochner technique.

Notations under affine connections {#subsec Ricci}
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As in the introduction, a two parameter family of [*torsion free*]{} affine connections $D^{\a, \g}$ is defined on $M$ for $\a,  \g\in \mathbb{R}$: $$\begin{aligned}
D^{\a,\g}_XY=%&\nb_XY+\a V^{-1}dV(X)Y+\a V^{-1}dV(Y)X +\g \bar g(X,Y)V^{-1}\nb V\nonumber
\nb_XY+\a du(X)Y+\a du(Y)X +\g \bar g(X,Y)\nb u.\end{aligned}$$ One checks directly that $D^{\a,\g}$ is torsion-free. For a general affine connection, we adapt the following convention of Ricci curvature.

\[defi Ricci\] Given an affine connection $D$, for any vector fields $X,Y$, we define the Ricci curvature as $$\begin{aligned}
       Ric^D(X,Y)=\o^i\left(R^D(e_i,X)Y\right),
   \end{aligned}$$ where $\{e_i\}$ is a local frame of the tangent bundle, $\{\o^i\}$ is the dual $1$-form of $\{e_i\}$ and the Riemann curvature operator $R^D$ is defined as $$\begin{aligned}
       R^D(X,Y)Z= D_{X}D_YZ-D_YD_{X}Z-D_{[X,Y]}Z.
   \end{aligned}$$

In the case we have a Riemannian metric $\bar g$, the Ricci curvature $Ric^D$ of $D$ can also be interpreted as $$\begin{aligned}
       Ric^D(X,Y)=\<R^D(e_i,X)Y, e_i\>,   \end{aligned}$$ where $\{e_i\}$ is an orthonormal frame of the tangent bundle.

By direct computation, we derive the following representation of $Ric^D$ for $D=D^{\a,\g}$ in terms of Levi-Civita connection $\bar\n$.

\[thm Ricci\] The Ricci curvature $Ric^D$ of $D=D^{\a,\g}$ satisfies the following identity: $$\begin{aligned}
       Ric^D=&Ric-[(n-1)\a+\g]\bar\nabla^2 u+[(n-1)\a^2-\g^2]du\otimes du\nonumber\\
        &+\left\{\g \bar\De u +\g[(n-1)\a+\g]|\nb u|^2\right\}\bar 