Singularities and normal forms of generic 2-distributions on 3-manifolds

B. Jakubczyk; M. Zhitomirskiĭ

Studia Mathematica (1995)

  • Volume: 113, Issue: 3, page 223-248
  • ISSN: 0039-3223

Abstract

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We give a complete classification of germs of generic 2-distributions on 3-manifolds. By a 2-distribution we mean either a module generated by two vector fields (at singular points its dimension decreases) or a Pfaff equation, i.e. a module generated by a differential 1-form (at singular points the dimension of its kernel increases).

How to cite

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Jakubczyk, B., and Zhitomirskiĭ, M.. "Singularities and normal forms of generic 2-distributions on 3-manifolds." Studia Mathematica 113.3 (1995): 223-248. <http://eudml.org/doc/216172>.

@article{Jakubczyk1995,
abstract = {We give a complete classification of germs of generic 2-distributions on 3-manifolds. By a 2-distribution we mean either a module generated by two vector fields (at singular points its dimension decreases) or a Pfaff equation, i.e. a module generated by a differential 1-form (at singular points the dimension of its kernel increases).},
author = {Jakubczyk, B., Zhitomirskiĭ, M.},
journal = {Studia Mathematica},
keywords = {2-distributions; normal forms},
language = {eng},
number = {3},
pages = {223-248},
title = {Singularities and normal forms of generic 2-distributions on 3-manifolds},
url = {http://eudml.org/doc/216172},
volume = {113},
year = {1995},
}

TY - JOUR
AU - Jakubczyk, B.
AU - Zhitomirskiĭ, M.
TI - Singularities and normal forms of generic 2-distributions on 3-manifolds
JO - Studia Mathematica
PY - 1995
VL - 113
IS - 3
SP - 223
EP - 248
AB - We give a complete classification of germs of generic 2-distributions on 3-manifolds. By a 2-distribution we mean either a module generated by two vector fields (at singular points its dimension decreases) or a Pfaff equation, i.e. a module generated by a differential 1-form (at singular points the dimension of its kernel increases).
LA - eng
KW - 2-distributions; normal forms
UR - http://eudml.org/doc/216172
ER -

References

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  1. [A] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1978. Zbl0386.70001
  2. [AI] V. I. Arnold and Yu. S. Il'yashenko, Ordinary Differential Equations, in: Modern Problems in Mathematics, Dynamical Systems 1, Springer, Berlin 1985. 
  3. [AVG] V. I. Arnold, A. N. Varchenko and S. M. Gusein-Zade, Singularities of Differentiable Maps, Vol. 1, Nauka, Moscow, 1982 (in Russian); English transl.: Birkhäuser, 1985. 
  4. [B] G. R. Belitskiĭ, Smooth equivalence of germs of vector fields with one zero eigenvalue or a pair of purely imaginary eigenvalues, Funktsional. Anal. i Prilozhen. 20 (4) (1986), 1-8 (in Russian). Zbl0657.58027
  5. [GG] M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Springer, Berlin, 1973. Zbl0294.58004
  6. [GT] M. Golubitsky and D. Tischler, On the local stability of differential forms, Trans. Amer. Math. Soc. 223 (1976), 205-221. Zbl0339.58003
  7. [JP] B. Jakubczyk and F. Przytycki, Singularities of k-tuples of vector fields, Dissertationes Math. 213 (1984). Zbl0565.58007
  8. [L] V. V. Lychagin, Local classification of first order nonlinear partial differential equations, Uspekhi Mat. Nauk 30 (1) (1975), 101-171 (in Russian). 
  9. [M] J. Martinet, Sur les singularités des formes différentielles, Ann. Inst. Fourier (Grenoble) 20 (1) (1970), 95-178. Zbl0189.10001
  10. [MZ] P. Mormul and M. Zhitomirskiĭ, Modules of vector fields, differential forms and degenerations of differential systems, Trans. Amer. Math. Soc., to appear. Zbl0866.58003
  11. [P] F. Pelletier, Singularités d'ordre supérieur de 1-formes, 2-formes et équations de Pfaff, Publ. Math. IHES 61 (1985). Zbl0568.58001
  12. [R] R. Roussarie, Modules locaux de champs et de formes, Astérisque 30 (1975). Zbl0327.57017
  13. [Z1] M. Zhitomirskiĭ, Singularities and normal forms of odd-dimensional Pfaff equations, Funktsional. Anal. i Prilozhen. 23 (1) (1989), 70-71 (in Russian). 
  14. [Z2] M. Zhitomirskiĭ, Typical Singularities of Differential 1-forms and Pfaffian Equations, Transl. Math. Monographs 113, Amer. Math. Soc., Providence, 1992. 
  15. [Z3] M. Zhitomirskiĭ, Finitely determined 1-forms ω, ω 0 ≠ 0 , are exhausted by Darboux and Martinet models, Funktsional. Anal. i Prilozhen. 19 (1) (1985), 59-61 (in Russian). 

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