An integral formula for affine connections

TL;DR

This paper introduces a new family of affine connections, derives their Ricci curvature, and develops an integral Bochner technique that generalizes previous results and yields new geometric inequalities and eigenvalue estimates.

Contribution

It presents a novel 2-parameter family of affine connections and an integral Bochner technique applicable under generalized Ricci curvature conditions.

Findings

New proof for substatic manifold results

Derivation of geometric inequalities

Eigenvalue estimates under generalized Ricci conditions

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 \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.