Hurkyl -- your (deleted) post was helpful. I see now that I was having trouble with the notation in which a vector times a scalar field denotes the directional derivative in that direction. Then the identity does follow, as you pointed out.
Thanks
Well the original problem is to show that if f(P) is a scalar field such that f(P_0)=1, and A,B,C are vector fields, then
[\nabla_A,\nabla_B](f(P)C(P))-[\nabla_A,\nabla_B{A}](C(P))=[\nabla_{[A,B]}(C(P))
Unless I am doing something wrong, this immediately reduces to the above identity, which...
I am trying to solve an exercise from MTW Gravitation and the following issue has come up:
Let D denote uppercase delta (covariant derivative operator)
[ _ , _ ] denotes the commutator
f is a scalar field, and A and B are vector fields
Question:
Is it true that
[D_A,D_B]f = D_[A,B]f
?