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Two metrics $g_{1}$ and $g_{2}$ are conformally equivalent metrics if $g_{2}=e^{2\theta}g_{1}$ A vector field $X$ is called conformal if $L_{X}g=2\theta g$ where $L_{X}$ is the Lie derivative with respect to $X$.

Is there a relation between both concepts?

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Yes, and you answered this question elsewhere: the flow of a conformal vector field is a family of conformal maps. That is, pulling back $g $ under a flow map gives you a conformally equivalent metric. –  Post No Bulls Dec 25 '13 at 7:47

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