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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.

1. , , 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]: .
2. J-antilinear 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].
3. Nijenhuis-Frölicher bracket (differential concomitant) of the vector valued 1-form J with itself. [FN].
4. 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]:

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

6. Weyl tensor of the homogeneous PDE (geometrical structure) modeled on the affine complex space ; the group is the second Spencer cohomology group. [KL].

7. , where is any symmetric connection on M. [K1].
8. Let g be a compatible metric, i.e. is a 2-form. Then the Nijenhuis tensor can be found from the following formula, where is the Levi-Civita connection of g (hence symmetric, see 7) [KN]: .
9. Let be 2-form (not metric as in 8). Then we can define the tensor by the formula . In the case when we can divide and , where . [K2]
10. The second generator of the invariant tensor algebra describing the image of the projection of pseudoholomorphic jets. [K1].
11. The real part of the curvature of the distribution generated by the projector of the complexified space ; . Hence . [KN], [Ko].
12. The homomorphism for non-holonomic 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 vector-valued differential forms'' (I), Proc. Koninkl. Nederl. Akad. Wetensch., ser.A, 59, issue 3 (1956), 338-359.

[Hö]

L.Hörmander, ''The Frobenius-Nirenberg theorem'', Arkiv for Mathematik 5 (1964), 425-432.

[KL]

B.Kruglikov, V.Lychagin ''On equivalence of differential equations'', Acta et Commentationes Universitatis Tartuensis de Matematica, 3 (1999), 7-29

[KN]

S.Kobayashi, K.Nomizu ''Foundations of Differential Geometry'' II, Wiley-Interscience (1969).

[Ko]

J.J.Kohn, ''Harmonic integrals on strongly pseudo-convex manifolds'' (I), Ann. of Math. 78 (1963), 206-213.

[K1]

B.Kruglikov ''Nijenhuis tensors and obstructions for pseudoholomorphic mapping constructions'', Mathematical Notes 63, issue 4 (1998), 541-561.

[K2]

B.Kruglikov, ''Some classificational problems in four dimensional geometry: distributions, almost complex structures and Monge-Ampere equations'', Math. Sbornik, 189, no. 11 (1998), 61-74.

[Li]

A.Lichnerowicz ''Theorie globale des connexions et des groupes d'holonomie'', Roma, Edizioni Cremonese (1955).

[LR]

V.Lychagin, V.Rubtsov ''Non-holonomic filtrations: Algebraic and geometric aspects of non-integrability'', Geometry in Partial differential equations, Ed. A.Prastaro, Th.M.Rassias (1994), 189-214.

[NN]

A.Newlander, L.Nirenberg ''Complex analytic coordinates in almost-complex manifolds'', Ann. Math. 65, ser. 2, issue 3 (1957), 391-404.

[NW]

A.Nijenhuis, W.Wolf ''Some integration problems in almost-complex and complex manifolds'', Ann. Math. 77 (1963), 424-489.

[St]

S. Sternberg ''Lectures on differential geometry'', Prentice-Hall, New Jersey (1964).

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