Let $(M,\rho)$ be a symplectic $2$-dimensional manifold, and let $J$ be a compatible complex structure on $M$, i.e. the symmetric $(0,2)$-tensor $$g(*,*) = \rho(*,J*)$$ is a Riemannian metric. Denote by $\nabla$ the Levi-Civita connection. Now let $V\in\Gamma(TM)$ be a vector field (with some further assumptions which are probably not needed). I would like to compute $$\left.\frac{d}{dt}\right|_{t=0}\left((d\varphi_V^{-t})_{\varphi_V^t(m)}J(\varphi_V^t(m))(d\varphi_V^t)_m\right),$$ where the map in the brackets is a map $T_mM\to T_mM$. Here $\varphi_V^t$ denotes the flow of $V$. If my computations are correct, I get the expression $$-\nabla_{J*}V + (\nabla_VJ)(*) + J(\nabla_*V)\tag{1},$$ which is again an endomorphism of $TM$. I would like to further simplify this expression. This would be easy if we had $$\nabla_{J*}V = J(\nabla_*V),$$ but I'm afraid that this is not true (at least without some more assumptions/structure). The only results I have about relations between $\nabla$ and $J$ are $$J(\nabla_*J) = (\nabla_*J)J\quad\mathrm{and}\quad\nabla_{J*}J = -J(\nabla_*J)$$ (see e.g. lemma 4.1.14 in this book), but I haven't been able to get anywhere using them.
My questions are:
- Is my computation $(1)$ correct?
- Is it possible to further simplify the expression $(1)$?
If you think that further assumptions are needed to solve the problem, please let me know.