$\a =\frac{1}{n-1}$ and $\g=0$ was proved in [@WY].

We have $D_X (V^\tau d\O) =0$ for any smooth vector field $X$.

Given an arbitrary vector field $W$ with compact support in $\O$. From Lemma \[weight\], $$\int_\O W^i D_i (V^\tau d\O)= -\int_\O  V^\tau  D_iW^i d\O =-\int_\O \nb_i(V^\tau W^i) d\O=0.$$ Since $W$ is arbitrary, we conclude that $D_i (V^\tau d\O)=0$ for any $i$.

Choosing $X=\nb^D \phi$ in , we obtain $$\begin{aligned}
   & D_i\Big((\nb^D \phi)^j D_j((\nb^D \phi)^i)-(\nb^D \phi)^i D_j((\nb^D \phi)^j)\Big)\nonumber \\=&|\bn^{2,D}\phi |_{\bar g}^2-|\bd^D \phi|^2+Ric^{D}(\bar\nabla^D\phi,\bar\nabla^D\phi).
    \label{Bochner equ 1'}\end{aligned}$$

Multiplying with $V^\tau$ and integrating over a bounded domain $\O\subset M$, we have $$\begin{aligned}
        &\di\int_{\O}V^\tau D_i\Big((\nb^D \phi)^j D_j((\nb^D \phi)^i)-(\nb^D \phi)^i D_j((\nb^D \phi)^j)\Big) d\O\nonumber\\
        =&\di\int_{\O}V^\tau \left[|\bn^{2,D}\phi |_{\bar g}^2-|\bd^D \phi|^2\right]+ V^\tau Ric^{D}(\bar\nabla^D\phi,\bar\nabla^D\phi) 
 d\O.
        \label{integral formula}
    \end{aligned}$$

Applying (\[div\]) and the Stokes’ theorem on , we obtain the Reilly type integral formula.

\[theo RF\] Let $(M^n, \bar g, V=e^u)$ be an $n$-dimensional Riemannian triple and $\a,\g\in \mathbb{R}$. Let $D=D^{\a,\g}$ be the affine connection defined as in (\[defi D\]) and $\tau=(n+1)\a+\g$. Let $\phi$ be a smooth function on a bounded domain $\O\subset M$ with smooth boundary $\S$. Then the following integral formula holds: $$\begin{aligned}
        &\di\int_{\O}V^\tau \left[\left|\bd^D \phi\right|^2-\left|\bn^{2,D} \phi\right|_{\bar g}^2 \right] -V^\tau Ric^{D}( \nb^D \phi, \nb^D \phi) d\O \\ =&\di\int_{\S}V^{\tau}\left[H^D\<\bn^D \phi,\nu\>^2+\left(h-\g u_\nu 