- #1

thecommexokid

- 70

- 2

## Homework Statement

Given a manifold

*M*with metric

*g*and associated derivative operator ∇

^{ab}*, let*

_{a}*ξ*be a Killing vector on

^{ a}*M*. Prove that

*ξ*is an affine collineation for ∇

^{ a}*.*

_{a}## Homework Equations

**(**For a vector

*a*)*ξ*to be an affine collineation for a derivative operator ∇

^{ a}*, it must satisfy*

_{a}∇

*∇*_{a}_{b }*ξ*=^{ c}*R*._{mab}^{c }ξ^{ m}**(**We are told that

*b*)*ξ*is a Killing vector, so

^{ a}∇

_{(a }*ξ*_{ b)}= 0.**(**The action of the Riemann tensor on a vector is

*c*)*R*= –2 ∇

_{abc}^{d }ξ^{ c}_{[a}∇

_{b] }

*ξ*.

^{ d}## The Attempt at a Solution

I've gone down one blind alley after another. Below I reproduce what seemed like the most promising path I've explored so far. Can somebody help me with the next step, or suggest an alternative path that could be more fruitful?

Start with the right-hand side of

**(**:

*a*)*R*=

_{mab}^{c }ξ^{ m}*g*.

^{cp }R_{mabp }ξ^{ m}*g*=

^{cp }R_{mabp }ξ^{ m}*g*=

^{cp }R_{bpma }ξ^{ m}*g*.

^{cp }g_{aq }R_{bpm}^{q }ξ^{ m}**(**:

*c*)*g*= –2

^{cp }g_{aq }R_{bpm}^{q }ξ^{ m}*g*∇

^{cp }g_{aq }_{[b}∇

_{p] }

*ξ*= –2

^{ q}*g*∇

^{cp}_{[b}∇

_{p] }

*ξ*.

_{ a}–2

Now it's easy to use *g*∇^{cp}_{[b}∇_{p] }*ξ*= –_{ a}*g*(∇^{cp}_{b}∇_{p }*ξ*– ∇_{ a}_{p}∇_{b }*ξ*) ._{ a}**(**on the first of these two terms:

*b*)–

But I still have that second term sitting there! What do I do about it?*g*(∇^{cp}_{b}∇_{p }*ξ*– ∇_{ a}_{p}∇_{b }*ξ*) = +_{ a}*g*(∇^{cp}_{b}∇_{a }*ξ*+ ∇_{ p}_{p}∇_{b }*ξ*)._{ a}