Given a Riemannian manifold without boundary, in the previous note, we derived a Bochner-type formula for the conformal Killing operator . Precisely, we obtained
Today, we try to derive a similar formula for the operator assuming the manifold has boundary . Our starting point again is the Bochner formula for vector fields mentioned here, i.e.
Using this and the formula for that we derived here, we arrive at
which now yields
We now integrate both sides over . First, using our previous calculation, there holds
Second, as before, we know that
Hence, we have proved that
Finally, making use of the identity
and the definition of , that is to say
we eventually arrive at
This is the so-called the Bochner-type formula for the conformal Killing operator on manifolds with boundary.