Almost-complex map

Definition

Let $(M,J_M)$ and $(N,J_N)$ be smooth manifolds with almost-complex structures. A smooth map $f:M\to N$ is an almost-complex map, also called a $J$-holomorphic map or pseudoholomorphic map, if

Equivalently, every tangent map $df_p:T_pM\to T_{f(p)}N$ is complex-linear for the complex structures defined by $J_M$ and $J_N$.

Identity maps, constant maps, and composites of almost-complex maps are almost-complex. If an almost-complex map is a diffeomorphism, its inverse is also almost-complex. Thus almost-complex manifolds and almost-complex maps form a category.

When both structures are integrable, the intertwining equation is equivalent to $f$ being a holomorphic map. For nonintegrable structures it remains meaningful, notably for pseudoholomorphic curves in symplectic geometry.