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This is from Hughston and Tod, Ex 8.5. Please give me an idea how to start with this.

If $\mathcal{L}_V g_{ij} = 2\phi g_{ij}$ then prove that $$ \mathcal{L}_V\Gamma^{i}_{jk} = \phi_{,j}\delta^{i}_{k} + \phi_{,k}\delta^{i}_{j} - g_{jk}\phi^{,i}. $$

I was thinking we could expand the connection in terms of the metric and then use Liebniz but I don't think the Lie derivative would commute past the derivative on the metric.

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