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Let $X$ be a complex manifold with a Hermitian metric. Is there a "complex" analogue of geodesics on $X$ which is of any interest? For example, is anything known about holomorphic maps $f : \mathbb C \rightarrow X$ satisfying $\nabla_{df/dz} (df/dz) = 0$ (which, naively, looks like a reasonable complex analogue of the geodesic condition)?

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