TITLE: Where does the notion of pseudoholomorphic curve come from?
QUESTION [8 upvotes]: I wonder, why we consider the notion of pseudoholomorphic curve: By definition a pseudoholomorphic curve in an almost complex manifold $X$ is a smooth map $f: C \rightarrow X$
from a Riemann surface $C$ into $X$ such that $df \circ J=J \circ df$ for the respective almost complex structure $J$. Why does it make sense to look at this definition? Is it just because the Cauchy-Riemann differential equations $\frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}$, $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$
are invariant under the symmetry $x \rightarrow y$, $y \rightarrow -x$ and $u \rightarrow v$, $v \rightarrow -u$ and therefore we have that this relation holds for a holomorphic map $f$ or is there any other reason?

REPLY [2 votes]: The work of Floer in proving the Arnold conjecture (at first in the monotone setting) interpreted pseudoholomorphic curves as a generalization of the flowlines in Morse theory.  In analogy with Morse homology, a count of pseudoholomorphic curves provides the differential in what has become known as a Floer homology.
This work predated Gromov's paper and has been seminal in its own way, spawning a variety of Floer homologies (such as the popular Heegaard-Floer homology) which provide strong invariants in low-dimensional topology.
If you're willing to accept that symplectic manifolds are reasonable things to study, then Floer's approach to the Arnold conjecture shows pseudoholomorphic curves arising very naturally.  Salamon has a nice paper explaining this carefully http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf .