The Bochner formula for a gradient vector field is a important tool in geometric analysis which basically says that

for any function . Whenever , we have a formula for as follows

Today, we discuss a variant of it called the Bochner formula for vector fields. I found this identity in a recent preprint of Li Ma [here].

Theorem. Let be a Riemannian manifold of dimension . Let be a smooth vector field on . Then we havewhere is the Lie derivative of the vector field with respect to underlying metric .