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## Homework Statement

Starting from

[itex]

-\xi^c(\Gamma^d_{ca}g_{db}+\Gamma^d_{cb}g_{ad})+\partial_b\xi^dg_{ad}+\partial_a\xi^cg_{cb}=0

[/itex]

I need to obtain the Killing equations, i.e.

[itex]

\bigtriangledown_b\xi_a+\bigtriangledown_a\xi_b=0

[/itex]

## Homework Equations

## The Attempt at a Solution

Working backwards...

Rewriting the covariant derivative in terms of the partial derivative gives

[itex]

\bigtriangledown_b\xi_a+\bigtriangledown_a\xi_b=\partial_a\xi_b+\partial_b\xi_a-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c=0

[/itex]

Lowering the vector in the partial derivatives gives...

[itex]

\partial_a\xi_b+\partial_b\xi_a-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c=-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c+\partial_b\xi^dg_{ad}+\partial_a\xi^cg_{cb}=0

[/itex]

I don't however know how to go from

[itex]

-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c[/itex]

to

[itex]

-\xi^c(\Gamma^d_{ca}g_{db}+\Gamma^d_{cb}g_{ad}[/itex])

Can someone help?