= Torsion-free Chern tangent connection characterizes a Kähler metric
{title2=$T_{\nabla^C}=0\iff d\omega=0$}
For a <Hermitian metric> on a <complex manifold>, the <Chern connection> on its <holomorphic tangent bundle> corresponds to a real metric-compatible connection preserving the <almost complex structure>. Its mixed torsion vanishes, and in holomorphic coordinates its remaining coefficients are $\Gamma^k_{ij}=h^{k\bar l}\partial_i h_{j\bar l}$. Symmetry in $i,j$ is equivalent to $\partial\omega=0$; its conjugate gives $d\omega=0$. Thus zero torsion is exactly the <Kähler metric> condition, and then uniqueness identifies this connection with the <Levi-Civita connection>.
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