In this note, I summary several useful integral identities on Riemannian manifolds with boundary.
- Suppose that is a function and is a -form, then
To prove this, we write everything in local coordinates as follows
- Using the previous identity, we can prove the following
where is again a vector field on . To prove this, we simply apply the previous identity with replaced by to get the desired result.
- Now we have a look-like integration by parts for scalar functions, actually, for any vector field over , we have
To prove this, we write
- We end this note by talking the following identity which can be thought of a generalization of the third identity. We prove
where is an -tensor, is an -tensor, and is a vector field given by
The proof is straightforward as one can easily see that
In view of the divergence theorem (or the Stokes theorem), we can further write
Let us go back to the third identity, we shall use and to get
where is given by
From this, we obtain the desired result through