You may also be amused by Bérard's article "From vanishing theorems to estimating theorems: the Bochner technique revisited", 
[Link](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-19/issue-2/From-vanishing-theorems-to-estimating-theorems--the-Bochner-technique/bams/1183554720.full)