Boris Kruglikov.
  "Nijenhuis tensors and obstructions for
  pseudoholomorphic mapping constructions".

  Mathematical Notes, v.63, no.4, 541--561 (1998).

We study obstructions for formal morphisms in the category of almost
complex manifolds. Such obstructions occur when the dimension of the
manifold-source is greater than 2. It turns out that the only
obstruction for 1-level lift in jets of the mappings is the condition of
Nijenhuis tensor jet preservation. But if we study the algebra of tensor
invariants, i.e. the projection of all invariants to the level of 1st
jets, this object turns to be more complicated. In particular it properly
contains the algebra, which is generated by (1,1)-tensor of complex
multiplication (almost complex structure) and by (2,1)-tensor of Nijenhuis
by means of Frolicher-Nijenhuis bracket. We also find the criterion of
local equivalence for almost complex manifolds.