  In this article, we introduce a $2$-parameter family of affine connections
and derive the Ricci curvature. We first establish an integral Bochner
technique. On one hand, this technique yields a new proof to our recent work in
\cite{LX} for substatic manifolds. On the other hand, this technique leads to
various geometric inequalities and eigenvalue estimates under a much more
general Ricci curvature conditions. The new Ricci curvature condition
interpolates between static Ricci tensor and $1$-Bakry-Emery Ricci, and also
includes the conformal Ricci as an intermediate case.
