 Solution

For a Riemann surface $(\Sigma ,j)$ and an almost complex manifold $(M,J)$, a J-holomorphic curve is a smooth map $u:\Sigma \to M$ satisfyingThus its differential is complex-linear at every point. In oriented local coordinates $s,t$ with $j{\partial }_s={\partial }_t$, this is $u_t=J{u}_s$, equivalently $u_s+J{u}_t=0$. No integrability of the target almost complex structure is required. Constant maps satisfy the definition; some usages reserve the word curve for nonconstant maps, so that restriction should be stated separately when intended.