Sur les singularités des formes différentielles. (French) Zbl 0189.10001

In this paper, the singularities of exterior differential forms on a manifold are studied from a view-point analogous to the classical theory of singularities of differentiable mappings. The main invariants considered are the rank and the class (in E. Cartan’s sense) of an exterior differential form. The generic behaviour of these invariants is studied by means of the transversality theorems. For instance, the set of points of a \(n\)-dimensional manifold, where the class of a one-form is \(n-c\), is generically a regular submanifold of codimension \({c(c+1)\over 2}\). Higher order singularities are studied in some detail in the case of low dimensions (1-forms in 3 dimensions and 2-forms in 4 dimensions). Then is proved the existence of models for the simplest singularities of 1-forms and closed 2-forms.
Reviewer: J. Martinet


58A15 Exterior differential systems (Cartan theory)
58K99 Theory of singularities and catastrophe theory
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[1] R. ABRAHAM and J. ROBBIN, Transversal mappings and flows. Benjamin, New-York, 1967. · Zbl 0171.44404
[2] N. BOURBAKI, Algèbre — Chap. 3, Algèbre multilinéaire. Hermann, Paris 1958.
[3] E. CARTAN, LES systèmes différentiels extérieurs et leurs applications géométriques, Hermann, Paris 1945. · Zbl 0063.00734
[4] S.S. CHERN, The geometry of G-structures. Bull. Amer. Math. Soc., 72, 1966, 167-219. · Zbl 0136.17804
[5] C. GODBILLON, Géométrie différentielle et mécanique analytique, Hermann, Paris 1969. · Zbl 0174.24602
[6] E. GOURSAT, Leçons sur le problème de Pfaff, Hermann, Paris 1922. · JFM 48.0538.01
[7] John W. GRAY, Some global properties of contact structures, Ann. Math. 69, 2, 1959, 421-450. · Zbl 0092.39301
[8] A. HAEFLIGER et A. KOSINSKI, Un théorème de thom sur LES singularités des applications différentiables, Séminaire H. Cartan, Ecole Normale Sup., Paris, Exposé 8, 1956/1957. · Zbl 0178.26604
[9] H.I. LEVINE, Singularities of differentiable mappings, Bonn, 1959. · Zbl 0216.45803
[10] P. LIBERMANN, Forme canonique d’une forme différentielle extérieure quadratique fermée, Bull. Ac. Roy. Belg. Cl. Sc. (5), 39, 1953, 846-850. · Zbl 0051.32601
[11] Jean MARTINET, Classe et points critiques d’une forme de Pfaff, C.R. Acad. Sc., Paris, t. 264, 1967, 97-99. · Zbl 0168.44401
[12] Jean MARTINET, Sur quelques singularités de formes différentielles admettant des modèles locaux, C.R. Acad. Sc., Paris, t. 266, 1968, 1 246-48. · Zbl 0162.54002
[13] J. MATHER, Stability of C∞-mappings V. Transversality (à paraître). · Zbl 0286.58006
[14] J. MOSER, On the volume elements on a manifold, Bull. Am. Math. Soc., 1965, 286-294. · Zbl 0141.19407
[15] R. THOM, LES singularités des applications différentiables. Ann. Inst. Fourier, Grenoble, VI, 1956, 43-87. · Zbl 0075.32104
[16] R. THOM, Un lemme sur LES applications différentiables. Bol. Soc. Mat. Mexic, 2a Serie, t. 1, 1956, 59-71. · Zbl 0075.32201
[17] H. WHITNEY, On singularities of mappings of Euclidean spaces, I. Mappings of the plane into the plane. Ann. Math. 62, 3, 1955, 374-410. · Zbl 0068.37101
[18] S. STERNBERG, Lectures on differential geometry, Prentice-Hall édit. U.S.A., 1964. · Zbl 0129.13102
[19] H. WHITNEY, Analytic extensions for differentiable functions defined on closed sets. Trans. Amer. Math. Soc., t. 36, 1934, 63-89. · JFM 60.0217.01
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