Let $M$ be a Kähler manifold (in truth, I am only interested in $\Bbb C \Bbb P^n$). Is it possible to express the Christoffel coefficients of the Levi-Civita connection in terms of the coefficients of the symplectic form? I ask because I am curious to see how they look in Darboux coordinates.
I know that $\Gamma _{ij} ^k$ can be expressed in terms of $g_{ij}$ and that these, in turn, can be expressed in terms of $\Omega_{ij}$ and $J_i ^j$. The problem is that this is my first encounter with these and I do not know how to manipulate them. The relevant chapter from Kabayashi & Nomizu doesn't really help me here.
I have
$$\Gamma _{ij} ^k = \frac 1 2 g^{ka} (g_{ia,j} + g_{aj,i} - g_{ij,a})$$
and $g_{ij} = J_j ^a \Omega_{ia}$ (which implies $g^{ij} = - \Omega ^{ia} J_a ^j$),
so it seems that
$$\Gamma _{ij} ^k = - \frac 1 2 \Omega ^{kc} J_c ^a (J_{a,j} ^b \Omega_{ib} + J_a ^b \Omega_{ib,j} + J_{j,i} ^b \Omega_{ab} + J_j ^b \Omega_{ab,i} - J_{j,a} ^b \Omega_{ib} - J_j ^b \Omega_{ib,a}) .$$
In Darboux coordinates, the derivatives of the coefficients of $\Omega$ are $0$, so one gets the simpler form
$$\Gamma _{ij} ^k = - \frac 1 2 \Omega ^{kc} J_c ^a (J_{a,j} ^b \Omega_{ib} + J_{j,i} ^b \Omega_{ab} - J_{j,a} ^b \Omega_{ib}) .$$
What do I do now with the coefficients of $J$? In Darboux coordinates, are their derivatives $0$, in order to get $\Gamma _{ij} ^k = 0$?