Pseudoholomorphic curves
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1 Introduction
In this article, will denote an almost complex manifold of dimension .
Definition 1.1 (-holomorphic curve). Let be a Riemann surface with complex structure . A -holomorphic curve
is a smooth map satisfying . A -holomorphic map is called simple if it cannot be factored as where is a -holomorphic branched cover of degree strictly greater than 1. We will usually omit from the notation and speak of -holomorphic curves. The term pseudoholomorphic curve will be used to describe a -holomorphic curve when we do not want to specify .
Pseudoholomorphic curves provide a useful tool for studying symplectic manifolds.