Ricci identity and Killing vectors in differential geometry

[latentcorpse]

Hi I'm trying Q1 of this paper:
http://www.maths.cam.ac.uk/postgrad/mathiii/pastpapers/2005/Paper60.pdf
and have got to the bit where I need to show that $\xi_{b;ca}=-R_{bca}{}^d \xi_d$

Now I know that $R_{bca}{}^d \xi_d=R_{bcad} \xi^d = R_{adbc} \xi^d = \nabla_b \nabla_c \xi_a - \nabla_c \nabla_b \xi_a$

where I got that last equality by rearranging the Ricci identity $\nabla_c \nabla_d Z^a - \nabla_d \nabla_c Z^a = R^a{}_{bcd}Z^b$

So then we have $-R_{bca}{}^d \xi_d = \nabla_c \nabla_b \xi_a - \nabla_b \nabla_c \xi_a = -\nabla_c \nabla_a \xi_b + \nabla_b \nabla_a \xi_c$
where I used the Killing property on the first term to get a $\xi_b$ term like we are looking for but I can't quite manipulate it into the final answer. Can anybody see what I am doing wrong?And then in the last bit, can somebody help me to show that if the Killing vector and the first derivative vanish at a point then they vanish everywhere?

And what about the final part about how many linearly independent Killing vectors can there be? My notes say that a n dimensional spacetime is maximally symmetric if there exist $\frac{n(n+1)}{2}$ linearly independent Killing vectors. But I don't actually know whether this is relevant to the question at hand or not?

Thanks.

 

[dextercioby]

I would start with

$$R_{[ij|k]l} = 0$$

from which you get based on Ricci's identity in the absence of torsion

$$\nabla_{[k}\nabla_{j}\xi_{i]} = 0$$

from which you should get by expansion into the 6 possible terms and regrouping based on

$$\nabla_{(i}\xi_{j)} = 0$$

an expression showing you the right way to get the final identity.

 

[latentcorpse]

I would start with

$$R_{[ij|k]l} = 0$$

from which you get based on Ricci's identity in the absence of torsion

$$\nabla_{[k}\nabla_{j}\xi_{i]} = 0$$

from which you should get by expansion into the 6 possible terms and regrouping based on

$$\nabla_{(i}\xi_{j)} = 0$$

an expression showing you the right way to get the final identity.

Thanks for your reply. What do you mean by $$R_{[ij|k]l} = 0$$? Is there supposed to be another vertical line in there somewhere to indicate one of the indices is not antisymmetrised?

 

[dextercioby]

No, sorry, the vertical line comes from the Youg tableau. So disregard it. I put it there from discussions on such tensors outside GR that have not left my mind yet.