---
abstract: '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 [@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-Émery Ricci, and also includes the conformal Ricci as an intermediate case.'
address:
- |
    Department of Mathematics\
    University of Alabama at Birmingham\
    Birmingham, AL 35294
- |
    School of Mathematical Sciences\
    Xiamen University\
    361005, Xiamen, P.R. China
author:
- Junfang Li and Chao Xia
title: An integral formula for affine connections
---

[^1]

Introduction
============

The classical Reilly formula is actually an integral Bochner formula for gradient vector fields on manifolds with boundary in references. It has been proven to be a quite useful tool in differential geometry.

Motivated by a work of Qiu and the second author [@QX], we have established a generalized Reilly type formula in previous work [@LX]. Such a generalization enabled us to prove a Heintze-Karcher-Ros-Brendle type inequality under 