# Lie derivative $\mathcal{L}_XJ(Y)$ with endomorphism $J$

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?

$$L_X(J(Y))$$ 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.
• 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. Nov 25 at 12:28