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Pseudo holomorphic curves in symplectic manifolds. (English) Zbl 0592.53025
Let $$V$$ be a smooth manifold with an almost complex structure $$J$$, that is a $$C^{\infty}$$-field of complex linear structures in the tangent spaces $$T_ v(V)$$, $$v\in V$$. If $$\text{dim }V=2$$, then $$(V,J)$$ is called a Riemann surface. A smooth map between two such manifolds, say $$f: (V',J')\to (V,J)$$, is called (pseudo) holomorphic if the differential $$D_ f: T(V')\to T(V)$$ is a complex linear map for the structures J’ and J. A parametrized (pseudoholomorphic) J-curve in an almost complex manifold (V,J) is a holomorphic map of a Riemann surface into $$V$$, say $$f: (S,J')\to (V,J)$$. The image $$C\in f(S)\subset V$$ is called a (non-parametrized) $$J$$-curve in $$V$$. A curve $$C\subset V$$ is called closed if it can be (holomorphically) parametrized by a closed surface $$S$$. $$C$$ is called regular if there is a parametrization $$f: S\to V$$ which is a smooth proper embedding. A curve is called rational if one can choose $$S$$ diffeomorphic to the sphere $$S^ 2.$$
The aim of the paper is an extension of basic facts on curves in complex manifolds to the almost complex case. The main results concern the existence of such curves in the presence of an auxiliary symplectic structure on $$(V,J)$$.
Reviewer: S.S.Singh

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 32Q99 Complex manifolds 32H99 Holomorphic mappings and correspondences
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