# Differential Geometry: Lie derivative of tensor fields.

1. Nov 1, 2011

### B L

1. The problem statement, all variables and given/known data
Let M be a differentiable manifold. Let X and Y be two vector fields on M, and let t be a tensor field on M. Prove
$\mathcal{L}_{[X,Y]}t = \mathcal{L}_X\mathcal{L}_Yt -\mathcal{L}_Y\mathcal{L}_Xt$

2. Relevant equations
All is fair game, though presumably a coordinate-free description is superior to one in local coordinates.

3. The attempt at a solution
I've tried working it out in local coordinates, but, because we need to show that the statement holds for a general tensor field of any type, the algebra gets very hairy very quickly. Any suggestions are much appreciated.