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I was reading Andrei Moroianu's Kahler Geometry note,there is a formula involove Lie derivative and endowmorphism $J$,seems not appear in the standard textbook about differential geometry.That is :

Let $(M,J)$ be a complex manifold and $J$ is a endowmorphism of $TM$,we have the following formula $$\mathcal{L}_X J(Y) = \mathcal{L}_X(J(Y)) - J(\mathcal{L}_XY)$$

I have no idea how to prove the formula above,one idea may written it as contraction of some tensor?

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I guess It's the Lie derivative extended on tensors. You have to think like a Leibniz's rule and you can understand the definition.

$L_X(J(Y))$ is $X(J(Y))$, you derived the function $J(Y)$ along $X$. Next, $J(L_XY)$ is like "$Y$ derived along $X$ and $J$ not derived" and $Y$ derived along $X$ is $[Y,X]$, and so on.

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  • $\begingroup$ My question is to prove the formula above, I know what both side are $\endgroup$
    – yi li
    Nov 25 at 10:14
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    $\begingroup$ It's a definition. Nothing to prove. It's the only operator that commutes with contractions. In any case wait for someone else if you dont'agree with me. $\endgroup$
    – Uoff
    Nov 25 at 10:14
  • $\begingroup$ thank you I got it $\endgroup$
    – yi li
    Nov 25 at 10:17
  • $\begingroup$ Whether this is just a definition depends a bit on the context. For applications, it will be important that, similar as for vector fields, $\mathcal L_XJ$ admits an interpretation in terms of the pullback of local flows of $X$. So for example, if $\mathcal L_XJ=0$, then local flows of $X$ are biholomorphisms. To see this, a proof is needed, but this basically only amounts to the fact that pullback along local diffeomorphisms commute with tensor products and contractions in an appropriate sense. $\endgroup$ Nov 25 at 12:28

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