g.
    \end{aligned}$$

We will prove Proposition \[thm Ricci\] in [**Appendix**]{}. As already mentioned in the introduction, the Ricci curvature $Ric^{D}$ of the new affine connection $D^{\a,\g}$ not only yields new Ricci curvature tensors, but also recovers some of known examples in the literature, which includes conformal Ricci tensor from conformal geometry, and static Ricci tensor raised from General Relativity and the $1$-Bakry-Éemery Ricci tensor from manifolds with density.

Next we explain the notations in .

\[notation\]

-   The $D$-gradient on $\O$ and $\S$ are defined respectively by $$\bn^D \phi:= V^{\g-\a}\bn \phi, \quad \n^D \phi:= V^{\g-\a}\n \phi$$

-   The $D$-Hessian $\bn^{2,D} \phi$ and $D$-Laplacian $\bd^{D} \phi$ on $\O$ are defined respectively by $$\begin{aligned}
    \bn^{2,D} \phi&:=& D(V^{\g-\a} \nb \phi)\\&=& V^{\g-\a} \left[\bn^2 \phi+\g du\otimes d\phi+\g d\phi \otimes du+ \a \<\bn u, \bn \phi\>\bar g\right], \nonumber\end{aligned}$$ and $$\begin{aligned}
    \bd^{D} \phi:= \tr_{\bar g} (\bn^{2,D} \phi)= V^{\g-\a} \left[\bd \phi+(2\g+ n\a)\<\bn u, \bn \phi\>\right]. \nonumber\end{aligned}$$

We note that in the case $\a=1, \g=-1$, the $D$-gradient, the $D$-Hessian and the $D$-Laplacian are in consistence with the classical ones for conformal metric $e^{2u}\bar g$. By virtue of this, we believe Definition \[notation\] is natural for $D^{\a,\g}$.

Bochner technique for general affine connections {#sec Bochner}
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In this section, we 