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 .
We note that the term is understood as following: Since is a tensor of type , its Lie derivative with respect to , is also a tensor of type . As such, before taking the divergence, we take and this is an -tensor which is able to take divergence.
Proof. Fix . Choose local orthonormal frame , that means
at . Then we calculate everything at . Write
For an -tensor , its Lie derivative with respect to is given by
i.e., Lie derivatives preserve the type of tensors. Using this, we can calculate as follows
By definition, the fact that , and making use of normal coordinates, we have
Keep in mind that in the previous calculation we have used for any smooth function and
Similarly, we also have
On the other hand
Combining all above quantities we have
which is the same as our formula.
As can be seen, the normal coordinates have been used several times to avoid massive calculation, for example, we have
The above identity has the following useful consequence: If is closed then by integrating both sides, we obtain
To see this, we write thanks to the fact that is a function, then we use the divergence theorem to get
Further calculation also shows that