This page helps you to understand me better

 The same page as pdf, jpg.

Dozen definitions of the Nijenhuis tensor tex2html_wrap_inline110 of an almost complex structure tex2html_wrap_inline112 .

This tensor is an obstruction for an almost complex structure to origin from the complex structure.


  1. tex2html_wrap_inline114 , tex2html_wrap_inline116 , where the right hand side is calculated for arbitrary vector fields X,Y with the given values tex2html_wrap_inline120 at the point tex2html_wrap_inline122 . In coordinates it has the formula [NN], [NW]: tex2html_wrap_inline131 .
  2. J-antilinear by each argument part of the torsion of any almost complex connection tex2html_wrap_inline128 (i.e. such a connection that tex2html_wrap_inline130 ) tex2html_wrap_inline132 . In other words, tex2html_wrap_inline141 . There are connections tex2html_wrap_inline128 called minimal such that tex2html_wrap_inline138 . [Li].
  3. Nijenhuis-Frölicher bracket (differential concomitant) tex2html_wrap_inline147 of the vector valued 1-form J with itself. [FN].
  4. Let tex2html_wrap_inline142 be the component of the de Rham differential. The Nijenhuis tensor is the only obstruction for the Dolbeault sequence to be a complex [Hö]: tex2html_wrap_inline150
  5. Structure function of the first order tex2html_wrap_inline146 for the G-structure with tex2html_wrap_inline150 associated with the almost complex structure J. [St].

  6. Weyl tensor tex2html_wrap_inline154 of the homogeneous PDE (geometrical structure) modeled on the affine complex space tex2html_wrap_inline156 ; the group tex2html_wrap_inline158 is the second Spencer cohomology group. [KL].

  7. tex2html_wrap_inline167 , where tex2html_wrap_inline128 is any symmetric connection on M. [K1].
  8. Let g be a compatible metric, i.e. tex2html_wrap_inline168 is a 2-form. Then the Nijenhuis tensor can be found from the following formula, where tex2html_wrap_inline128 is the Levi-Civita connection of g (hence symmetric, see 7) [KN]: tex2html_wrap_inline180 .
  9. Let tex2html_wrap_inline176 be 2-form (not metric as in 8). Then we can define the tensor by the formula tex2html_wrap_inline178 . In the case when tex2html_wrap_inline180 we can divide tex2html_wrap_inline182 and tex2html_wrap_inline190 , where tex2html_wrap_inline186 . [K2]
  10. The second generator of the invariant tensor algebra tex2html_wrap_inline188 describing the image of the projection tex2html_wrap_inline197 of pseudoholomorphic jets. [K1].
  11. The real part of the curvature of the distribution tex2html_wrap_inline192 generated by the projector of the complexified space tex2html_wrap_inline201 ; tex2html_wrap_inline196 . Hence tex2html_wrap_inline198 . [KN], [Ko].
  12. The homomorphism tex2html_wrap_inline200 for non-holonomic filtration of the projective module determined by the module tex2html_wrap_inline202 . In the almost complex case tex2html_wrap_inline204 . [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), 391-404.

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

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

To see the next page click here

Back to the main page

../Images/wb02.gif (290 bytes)