Given a $(1,0)$ smooth tangent vector field $L$ in $\Bbb C^n$, that is $$ L=\sum_{j=1}^na_j\frac{\partial}{\partial z_j} $$ where $a_j$'s are complex valued functions, differentiable in the real sense, what does $$ \nabla_L $$ mean? Someone told me that is an operator which, on gradients $$ \nabla r=\sum_{j=1}^n\partial_{z_j}r\cdot\partial_{z_j}+\partial_{\overline{z_j}}r\cdot\partial_{\overline{z_j}} $$ behaves as follows $$ \nabla_L\nabla r =\sum_{j=1}^nL\left(\partial_{z_j}r\right)\cdot\partial_{z_j}+L\left(\partial_{\overline{z_j}}r\right)\cdot\partial_{\overline{z_j}} $$ that is, it acts on the coefficients, but I would like to understand something more about it. Can someone give me some reference, some hint?


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