Assume we have $X$ a complex manifold and $Y = Y^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ and $Z = Z^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ two vector fields on $X$. Let $\nabla$ be the covariant derivative. What is the computation of $\nabla_{Z}Y$ (i.e. what is $\nabla_{\frac{\partial}{\partial z^{\beta}}}\frac{\partial}{\partial z^{\alpha}}$, i think its $0$ but why ?)?

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holomorphiccoordinates on the manifold, as I assume you mean, then you can choose a connection such thatin that single coordinate chart, this condition holds, but youcannotalways ensure this holds in every single coordinate chart. Because if it did, this would be a "flat" connection, and not all complex manifolds admit flat connections. – Spiro Karigiannis Jun 12 '11 at 13:32