# Homework Help: Killing vectors

1. Jan 9, 2010

### Altabeh

I was wondering if you could help me with the proof of the following theorem.

If $$A^{{\mu }}$$ and $$B^{{\mu }}$$ are Killing vectors, then so is their commutator $$C^{{\mu }}=[A,B]^{\mu}$$.

2. Jan 9, 2010

### diazona

What have you tried?

3. Jan 10, 2010

### Altabeh

Assuming $$g_{\mu \nu}$$ as our metric, I can write

$$g_{\mu \alpha}A^{\alpha}_{;\nu}=-g_{\alpha\nu}A^{\alpha}_{;\mu}$$ and
$$g_{\mu \alpha}B^{\alpha}_{;\nu}=-g_{\alpha\nu}B^{\alpha}_{;\mu}$$.

And I can correlate these two to each other, but I'm afraid about the commutator. I just need a clue as to how the commutator is written in terms of $$A^{\mu}$$ and $$B^{\mu}$$.