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Dozen definitions of the Nijenhuis tensor of an almost complex structure .
This tensor is an obstruction for an almost complex structure
to origin from the complex structure.

, ,
where the right hand side is calculated for arbitrary vector fields
X,Y with the given values at the point .
In coordinates it has the formula [NN], [NW]:
.

Jantilinear by each argument part of the torsion of any almost
complex connection (i.e. such a connection that )
. In other words,
.
There are connections called minimal such that .
[Li].
 NijenhuisFrölicher bracket (differential concomitant)
of the vector valued 1form J with itself. [FN].
 Let
be the component of the de Rham differential. The Nijenhuis tensor is the
only obstruction for the Dolbeault sequence to be a complex [Hö]:

Structure function of the first order for the Gstructure
with associated with the almost complex structure J.
[St].

Weyl tensor of
the homogeneous PDE (geometrical structure) modeled on the affine complex
space ; the group is the second
Spencer cohomology group. [KL].
 ,
where is any symmetric connection on M. [K1].
 Let g be a compatible metric, i.e. is a
2form. Then the Nijenhuis tensor can be found from the following formula,
where is the LeviCivita connection of g (hence symmetric, see
7) [KN]:
.
 Let be 2form (not metric as in 8). Then
we can define the tensor by the formula . In the case when we can divide and
,
where . [K2]
 The second generator of the invariant tensor algebra
describing the image of the projection
of pseudoholomorphic jets. [K1].
 The real part of the curvature of the distribution
generated by the projector of the complexified space
; . Hence
.
[KN], [Ko].
 The homomorphism for nonholonomic filtration of the projective module
determined by the module . In the almost complex case . [LR].
B.K.
References.
 [FN]
 A.Frolicher, A.Nijenhuis ''Theory of vectorvalued
differential forms'' (I), Proc. Koninkl. Nederl. Akad. Wetensch., ser.A,
59, issue 3 (1956), 338359.
 [Hö]
 L.Hörmander, ''The FrobeniusNirenberg theorem'',
Arkiv for Mathematik 5 (1964), 425432.
 [KL]
 B.Kruglikov, V.Lychagin ''On equivalence of
differential equations'', Acta et Commentationes Universitatis
Tartuensis de Matematica, 3 (1999), 729
 [KN]
 S.Kobayashi, K.Nomizu ''Foundations of Differential Geometry''
II, WileyInterscience (1969).
 [Ko]
 J.J.Kohn, ''Harmonic integrals on strongly pseudoconvex
manifolds'' (I), Ann. of Math. 78 (1963), 206213.
 [K1]
 B.Kruglikov ''Nijenhuis tensors and obstructions for
pseudoholomorphic mapping constructions'', Mathematical Notes 63,
issue 4 (1998), 541561.
 [K2]
 B.Kruglikov, ''Some classificational problems in four
dimensional geometry: distributions, almost complex structures and
MongeAmpere equations'', Math. Sbornik, 189, no. 11 (1998), 6174.
 [Li]
 A.Lichnerowicz ''Theorie globale des connexions et des
groupes d'holonomie'', Roma, Edizioni Cremonese (1955).
 [LR]
 V.Lychagin, V.Rubtsov ''Nonholonomic filtrations:
Algebraic and geometric aspects of nonintegrability'', Geometry in Partial
differential equations, Ed. A.Prastaro, Th.M.Rassias (1994), 189214.
 [NN]
 A.Newlander, L.Nirenberg ''Complex analytic coordinates in
almostcomplex manifolds'', Ann. Math. 65, ser. 2, issue 3 (1957),
391404.
 [NW]
 A.Nijenhuis, W.Wolf ''Some integration problems in
almostcomplex and complex manifolds'', Ann. Math. 77 (1963),
424489.
 [St]
 S. Sternberg ''Lectures on differential geometry'',
PrenticeHall, New Jersey (1964).
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