= Metric compatibility as parallelism of a Hermitian tensor
{title2=$H\in\Gamma((E\otimes\overline E)^*),\quad D_0H=0$}
With a <Hermitian form> linear in its first slot, $H(s\otimes\bar t)=h(s,t)$ is a complex-linear dual tensor. The original connection, its <conjugate connection>, the <tensor product connection> and the <dual connection> induce $D_0$. Its derivative is $(D_{0,V}H)(s\otimes\bar t)=Vh(s,t)-h(D_Vs,t)-h(s,D_Vt)$. Consequently $D_0H=0$ is exactly metric compatibility.
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