mean curvature.\
Formula reduces to Reilly’s original formula for $\bar g$ in the case $\a=\g=0$ and for $e^{2\a u}\bar g$ in the case $\a=-\g$. Moreover, when $\a=0, \g=1$, it reduces to the following

Let $(M^n, \bar g, V=e^u)$ be an $n$-dimensional Riemannian triple. Let $\phi$ be a smooth function on a bounded domain $\O\subset M$ with smooth boundary $\S$. We have $$\begin{aligned}
  \label{theo RF sub}
        &\di\int_{\O}V^3\Big[(\bar\De \phi+2\bar\n u\bar\n \phi)^2-|\bar\n_i\bar\n_j\phi+\bar\n_i u\bar\n\phi_j+\bar\n_j u\bar\n_i\phi|^2\Big]d\O\\
        =&\di\int_{\S}V^{3}\Big(H\phi_\nu^2+\left(h-u_\nu g\right)(\n\phi, \n\phi)-2 V^{-1}\<\n \phi, \n (V\phi_\nu)\>\Big)dA\nonumber\\
        &+\int_{\O}V^{3}(Ric-\frac{\nb^2 V}{V}+ \frac{\bar\De V}{V}\bar g)(\nb\phi,\nb\phi)d\O.\nonumber
          \end{aligned}$$

If we let $\phi=\frac{f}{V}$ in , then we recover Theorem 1.1 in [@LX] by a different method.

Let us illustrate the basic idea of the proof of Theorem \[theo RF0\]. It is well known that a local Bochner formula holds for general vector fields under an affine connection. Since the connection is in general not metric compatible, we have only a divergent form instead of the Laplacian of some function in the Bochner identity, see Proposition \[Bochner formula\]. Nevertheless, we are able integrate this local Bochner formula to get an integral formula. To achieve an effective Reilly type formula, there are two innovative ingredients with this method. First, we choose a right volume form, which is 