= Solution
For a <Riemann surface> $(\Sigma,j)$ and an <almost complex manifold> $(M,J)$, a <J-holomorphic curve> is a <smooth> map $u:\Sigma\to M$ satisfying
$$
\boxed{du\circ j=J\circ du,\qquad\bar\partial_Ju:=\frac12(du+J\circ du\circ j)=0.}
$$
Thus its differential is complex-linear at every point. In oriented local coordinates $s,t$ with $j\partial_s=\partial_t$, this is $u_t=Ju_s$, equivalently $u_s+Ju_t=0$. 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.
Back to article page
