positive smooth function on $M$, where $u$ is a smooth function on $M$. We call $(M, \bar g, V)$ a Riemannian triple.

For $\a, \g\in \mathbb{R}$, we define a $2$-parameter family of affine connections: given two vector fields $X, Y$ on $M$, let $$\begin{aligned}
D^{\a,\g}_XY=%&\nb_XY+\a V^{-1}dV(X)Y+\a V^{-1}dV(Y)X +\g \bar g(X,Y)V^{-1}\nb V\nonumber
\nb_XY+\a du(X)Y+\a du(Y)X +\g \bar g(X,Y)\nb u.
        \label{defi D}\end{aligned}$$ For simplicity, we often omit the superscript $\a,\g$ when it is clear in the context. The Ricci curvature under $D^{\a,\g}$ is (see Proposition \[thm Ricci\] below) $$\begin{aligned}
  \label{Ricci}
Ric^{D}:=&Ric-[(n-1)\a+\g]\bar\nabla^2 u+\big[(n-1)\a^2-\g^2\big]du\otimes du\nonumber\\
        &+\left[\g \bar\De u +(\g^2+(n-1)\a\g)|\nb u|^2\right]\bar g.
  \end{aligned}$$

There are two trivial cases among all $D^{\a,\g}$. One is the Levi-Civita connection for $\bar g$ when $\a=\g=0$, while the other is the Levi-Civita connection for the conformal metric $e^{2\a u} \bar g$ when $\a=-\g$. For other choices of $\a$ and $\g$, $D^{\a,\g}$ may not be a Levi-Civita connection for any Riemannian metric.

For the case $\a=0, \g=1$, one sees from that $$\begin{aligned}
Ric^{D^{0,1}}=  Ric-\frac{\bar \nabla^2 V}{V}+\frac{\bar \Delta V}{V}\bar g,\end{aligned}$$ where $Ric$ is the usual Ricci curvature for $\bar g$. We call $Ric^{D^{0,1}}$ [*static Ricci tensor*]{}. A Riemannian triple $(M, \bar g, V)$ satisfying $VRic^{D^{0,1}}=0$ is referred to a static manifold in 