In this notes, I want to summarize some properties of the so-called Conformal Killing Operator relative to the metric , say .
1. First, we start with its definition. Roughly speaking, the conformal killing operator is a generalization of the Killing operator relative to the metric . It maps any vector field on to some tensor of type . More precisely, in components, we have
where is a vector field on .
Immediately, one can check that is traceless as can be seen in the following
2. We now define the so-called Conformal Vector Laplacian associated to the metric . Basically, it is given by
In components, we have
3. Let us now determine the kernel of . By definition, one can easily see that . However, we shall prove that they are actually the same. For any vector field , we first have
where the Gauss-Ostrogradsky theorem has been used to get the last line. In view of the right-hand side integrand, we see that
where we have used the symmetry and the traceless property of . Therefore, we can write
Assume that , we then see that since the metric is positive definite.
By integration by parts, we have the following
for any vector fields and and is the outward normal vector field on . As above, we can write
4. Solutions to the Conformal Vector Poisson Equation. Let now discuss the existence and uniqueness of solutions to the conformal vector Poisson equation
where the vector field is already given. We shall prove that a necessary condition for the solution to exist is that the source must be orthogonal to any vector field in the kernel, in the sense that
This is clear as the following. First we know that
Because , we obtain the desired identity.
Clearly if admits no conformal Killing vector, the identity is trivial. To see why this condition also gives us a sufficient condition, we use the Fredholm theory.