review a Bochner identity for general affine connection and prove Theorem \[theo RF0\].

It is well known that under Levi-Civita connection $\nb$, the following Bochner formula holds: for a smooth vector field $X$ on $M$ with the property that $\nb X$ is symmetric, $$\begin{aligned}
\bar \De \frac12|X|^2=|\bar \n X|^2+ \bar \nabla_X (\div_{\bar g} X)+Ric(X, X),\end{aligned}$$ see e.g. Petersen [@Petersen] Proposition 33, page 207.

Under an affine connection, the following Ricci identity holds.

[**(Ricci identity)**]{}\[Ricci iden\] Under local coordinates $\{\P_i\}$, for any smooth vector field $X$, we have $$\begin{aligned}
        D_iD_jX^i=D_jD_iX^i+R^{D}_{jk}X^k.
        \label{Ricci identity}
    \end{aligned}$$

A vector filed $X$ can be viewed as a $(1,0)$-tensor field and we have $$\begin{aligned}
        D^2X(V,W)=&D_W(D_VX)-D_{D_WV}X,\nn
        D^2X(W,V)=&D_V(D_WX)-D_{D_VW}X.
        \label{Ricci identity proof equ 1}
    \end{aligned}$$ It follows that $$\begin{aligned}
        D^2X(W,V)-D^2X(V,W)=&R^{D}(V,W)X.
        \label{Ricci identity proof equ 2}
    \end{aligned}$$ Let $\{e_i\}$ be an local frame of the tangent bundle, $\{\o^i\}$ is the dual $1$-form of $\{e_i\}$. Then we have from $$\begin{aligned}
       \o^i\left(D^2X(e_j,e_i)\right)- \o^i\left(D^2X(e_i,e_j)\right)=&Ric^{D}(e_j, X).
    \end{aligned}$$ We finish the proof.

\[Bochner formula\] [**(Bochner formula)**]{} Let $D$ be an affine connection on $M$ and $Ric^D$ be the Ricci curvature of $D$. Let $X$ be a smooth vector field on $M$. Then we have $$\begin{aligned}
    \div^D (D_XX)=(DX)^t \cdot DX+d(\div^D X)(X)+Ric^{D}(X, X).
    \label{Bochner}\end{aligned}$$ where $div^D$ is the 