the literature, see e.g. [@Cor].

For the case $\a=\frac{1}{n-1}, \g=0$, one sees from that $$\begin{aligned}
Ric^{D^{\frac{1}{n-1},0}}=  Ric-\mathrm{\bar \nabla^2} u +\frac{1}{n-1}d u\otimes d u.\end{aligned}$$ This is in fact the [*$1$-Bakry-Émery Ricci tensor*]{} in the literature which was introduced by Bakry and Émery [@BaE]. The fact that this affine connection gives rise to the $1$-Bakry-Émery Ricci tensor has also been observed by Wylie-Yeroshkin [@WY] in their recent studies of manifolds with density.

The main result of this paper is the following Reilly type integral formula.

\[theo RF0\] 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: $$\begin{aligned}
         \label{formula RF 1}
        &\di\int_{\O}V^\tau \left[\left|\bd^D \phi\right|^2-\left|\bn^{2,D} \phi\right|_{\bar g}^2 \right] -V^\tau Ric^{D}( \nb^D \phi, \nb^D \phi) d\O \\ =&\di\int_{\S}V^{\tau}\left[H^D\<\bn^D \phi,\nu\>^2+\left(h-\g u_\nu g\right)(\n^D\phi, \n^D \phi)-2V^{-\g}\<\n^D \phi, \n^D(V^\g \phi_\nu)\>\right]dA.\nonumber
    \end{aligned}$$

The notations $\nb^D$, $\bn^{2,D}$ and $\bd^D$ play the role of gradient, Hessian and Laplacian with respect to $D$, the exact definition will be given in Definition \[notation\]. $H^D:=H+(n-1)\a u_\nu$ is the affine mean curvature, where $H$ is the usual 