In proposition 58 Chapter 1 in the excellent book O’Neill (1983), the author demonstrates that the Lie derivative of one vector field with respect to another is the same as the Lie bracket (of the two vector fields) although he calls the Lie bracket just bracket and does not define the Lie derivative preferring just to use its definition with giving it a name. The proof relies on a prior result where he shows a co-ordinate system at a point can be given to a vector field for which so that .
Here’s a proof seems clearer (to me at any rate) and avoids having to distinguish the case wehere the vector field is zero or non-zero. These notes give a similar proof but, strangely for undergraduate level, elide some of the details.
A Few Definitions
Let be a smooth mapping and let be a tensor with then define the pullback of by to be
For a tensor the pullback is defined to be .
Standard manipulations show that is a smooth (covariant) tensor field and that is -linear and that .
Let be a diffeomorphism and a vector field on we define the pullback of this field to be
Note that the pullback of a vector field only exists in the case where is a diffeomorphism; in contradistinction, in the case of pullbacks of purely covariant tensors, the pullback always exists.
For the proof below, we only need the pullback of functions and vector fields; the pullback for tensors with is purely to give a bit of context.
From O’Neill (1983) Chapter 1 Definition 20, let be a smooth mapping. Vector fields on and on are –related written if and only if .
The Alternative Proof
By Lemma 21 Chapter 1 of O’Neill (1983), and are -related if and only if .
Recalling that and since
we see that the fields and are -related: . Thus we can apply the Lemma.
Although we don’t need this, we can express the immediately above equivalence in a way similar to the rule for covariant tensors
First let’s calculate the Lie derivative of a function with respect to a vector field where is its flow
Analogously defining the Lie derivative of with respect to
Since we have
O’Neill, B. 1983. Semi-Riemannian Geometry with Applications to Relativity, 103. Pure and Applied Mathematics. Elsevier Science. https://books.google.com.au/books?id=CGk1eRSjFIIC.