Classification of differential (n — 1)-forms on an n-dimensional manifold with boundary

  • Wojciech Domitrz
Part of the Mathematics and Its Applications book series (MAIA, volume 350)


In this paper we classify differential n-forms and (n — 1)-forms on an n-dimentional manifold with boundary with respect to the equivalence defined by pullback via a diffeomorphism, which preserves the manifold together with its boundary. We present a complete list of the locally stable n-forms. We also classify all the stable (n — 1)-forms, provided a kernel of these forms meets the boundary transversally. We show that a 1-form on a 2-dimensional manifold, which does not satisfy the above condition, is not locally stable.


Vector Field Smooth Function Stable Mapping Symplectic Structure Smooth Hypersurface 
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  1. 1.
    V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko: Singularities of differentiable maps vol. I, Birkhäuser, Boston, 1985.zbMATHCrossRefGoogle Scholar
  2. 2.
    V.I. Arnold: Critical points of functions on a manifold with boundary, the simple Lie groups B k, C k and F k and singularities of evolutes, Uspekhi Matematicheskykh Nauk, 33(5), (1978), 91–105.Google Scholar
  3. 3.
    W. Domitrz, S. Janeczko: Normal forms of symplectic structures on the stratified spaces, to be published in Colloquium Mathematicum.Google Scholar
  4. 4.
    W. Domitrz, S. Janeczko: On Martinet’s singular symplectic structures, to be published in Banach Center Publications 1995.Google Scholar
  5. 5.
    M. Golubitsky, D. Tischler: A survey on the singularities and stability of differential forms, Asterisque, 59-60 (1978), 43–82.MathSciNetGoogle Scholar
  6. 6.
    M. Golubitsky, D. Tischler: An example of moduli for singular symplectic forms, Inventiones Math., 38 (1977), 219–225.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    M. Golubitsky, D. Tischler: On local stability of differential forms, Trans. A.M.S., 223 (1970), 205–221.MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Golubitsky, V. Guillemin: Stable mappings and their singularities, Springer-Verlag, 1973.Google Scholar
  9. 9.
    J. Martinet: Surles singularitesdés formes differentielles, Ann. Inst. Fourier (Grenoble), 20 (1970), 95–178.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    J. Martinet: Singularities of smooth functions and maps, Cambridge Univ. Press, Cambridge 1982.Google Scholar
  11. 11.
    J. Moser: On volume elements on manifolds, Trans. A.M.S., 120, 2 (1965), 280–296.CrossRefGoogle Scholar
  12. 12.
    R. Roussarie: Modèles locaux de champs et de formes, Astérisque, 30, (1975), 1–181.MathSciNetGoogle Scholar
  13. 13.
    M. Zhitomirskii: Typical singularities of differential 1-forms and Pfaffian equations, Translations of Mathematical Monographs, vol 113, A.M.S., Providence, RI, 1992.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Wojciech Domitrz
    • 1
  1. 1.Institute of MathematicsWarsaw University of TechnologyWarsawPoland

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