The standard definition of the lie derivative of X along Y is just(adsbygoogle = window.adsbygoogle || []).push({});

$$(*) \mathcal{L}_YX = \lim_{t\to 0} \frac{X_{\phi(t)} - \phi_{t*}X}{t}$$

where ##\phi_t## is the flow generated by Y. I.e. the limit of the difference between a pushforward of X along Y and X evaluated at a point ##\phi(t)## on the flow of Y. Other than this definition I have seen the expressions

$$(1) \mathcal{L}_Y X = [Y,X]$$

and

$$(2) \mathcal{L}_YX = \nabla_Y X - \nabla_X Y$$

for the Lie Derivative, but I have not seen it stated clearly when (1) and (2) holds. Surely (1) and (2) is equivalent when the connection ##\nabla## is torsion free. However is it generally true that (*) is equivalent with (1) or (2)? What conditions on ##\nabla## must be satisfied in order to identify (*) with (1) (or (2))?

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# Clarification on Lie Derivatives

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