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 have
where
is the Lie derivative of the vector field
with respect to underlying metric
.