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Does the following second covariant (in terms of Kahler geometry) derivative of Kahler potential vanish? \begin{equation} K_{ij}\equiv\nabla_i\nabla_j K=0, \end{equation} \begin{equation} K_{i^*j^*}\equiv\nabla_{i^*}\nabla_{j^*} K=0? \end{equation} Indices represent complex coordinates ($A_i$ and $\bar{A}_i$) on Kahler manifold.

I need this to show that \begin{equation} K_i\Box A_i=-K_{ij^*}\partial^m A_i\partial_m\bar{A}_j+\text{total derivative}, \end{equation} where partial derivatives and the box operator are taken WRT space-time coordinates (for which $A$ and $\bar{A}$ are scalar fields, of course).

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