Nijenhuis tensors and obstructions for pseudoholomorphic mapping constructions

Abstract

In this paper we present some approaches to classification of almost complex structures and to construction of local or formal pseudoholomorphic mapping from one almost complex manifold to another. The corresponding criteria are given in terms of Nijenhuis tensors and their generalizations. We deal with the prolongations of the Cauchy-Riemann equation in 1-jets of the mapping from one almost complex manifold to another. We give a criterion of the prolongation existence of k-th pseudoholomorphic jets to the (k+1)-th ones. As a consequence basing on the Kuranishi's approach we obtain formal normal forms of almost complex structures. As another consequence we get the formal part of Newlander-Nirenberg's theorem on integrability. We also introduce the notion and study the properties of the linear Nijenhuis tensor, which we apply to almost complex structures of general position. Two applications are exhibited: a classification of four-dimensional almost complex structures in terms of distributions and a characterization of the new class of structures which we call Lie almost complex structures.