Pseudoholomorphic curve

In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or J-holomorphic curve) is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann equation. Introduced in 1985 by Mikhail Gromov, pseudoholomorphic curves have since revolutionized the study of symplectic manifolds. In particular, they lead to the Gromov–Witten invariants and Floer homology, and play a prominent role in string theory.

Let ${\displaystyle X}$ be an almost complex manifold with almost complex structure ${\displaystyle J}$. Let ${\displaystyle C}$ be a smooth Riemann surface (also called a complex curve) with complex structure ${\displaystyle j}$. A pseudoholomorphic curve in ${\displaystyle X}$ is a map ${\displaystyle f:C\to X}$ that satisfies the Cauchy–Riemann equation

...
Wikipedia