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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]:
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 .
- Nijenhuis-Frölicher bracket (differential concomitant)
of the vector valued 1-form J with itself. [FN].
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 G-structure
with associated with the almost complex structure J.
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
2-form. Then the Nijenhuis tensor can be found from the following formula,
where is the Levi-Civita connection of g (hence symmetric, see
- 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]
- 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
- The homomorphism for non-holonomic filtration of the projective module
determined by the module . In the almost complex case . [LR].
- A.Frolicher, A.Nijenhuis ''Theory of vector-valued
differential forms'' (I), Proc. Koninkl. Nederl. Akad. Wetensch., ser.A,
59, issue 3 (1956), 338-359.
- L.Hörmander, ''The Frobenius-Nirenberg theorem'',
Arkiv for Mathematik 5 (1964), 425-432.
- B.Kruglikov, V.Lychagin ''On equivalence of
differential equations'', Acta et Commentationes Universitatis
Tartuensis de Matematica, 3 (1999), 7-29
- S.Kobayashi, K.Nomizu ''Foundations of Differential Geometry''
II, Wiley-Interscience (1969).
- J.J.Kohn, ''Harmonic integrals on strongly pseudo-convex
manifolds'' (I), Ann. of Math. 78 (1963), 206-213.
- B.Kruglikov ''Nijenhuis tensors and obstructions for
pseudoholomorphic mapping constructions'', Mathematical Notes 63,
issue 4 (1998), 541-561.
- 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.
- A.Lichnerowicz ''Theorie globale des connexions et des
groupes d'holonomie'', Roma, Edizioni Cremonese (1955).
- 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.
- A.Newlander, L.Nirenberg ''Complex analytic coordinates in
almost-complex manifolds'', Ann. Math. 65, ser. 2, issue 3 (1957),
- A.Nijenhuis, W.Wolf ''Some integration problems in
almost-complex and complex manifolds'', Ann. Math. 77 (1963),
- S. Sternberg ''Lectures on differential geometry'',
Prentice-Hall, New Jersey (1964).
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