Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The structure of the set of singular points of a codimension $ 1$ differential system on a $ 5$-manifold


Authors: P. Mormul and M. Ya. Zhitomirskiĭ
Journal: Trans. Amer. Math. Soc. 342 (1994), 619-629
MSC: Primary 58A17; Secondary 58A30, 58C27
DOI: https://doi.org/10.1090/S0002-9947-1994-1150017-9
MathSciNet review: 1150017
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Generic modules V of vector fields tangent to a 5-dimensional smooth manifold M, generated locally by four not necessarily linearly independent fields $ {X_1}$, $ {X_2}$, $ {X_3}$, $ {X_4}$, are considered. Denoting by $ \omega $ the 1-form $ {X_4}\lrcorner{X_3}\lrcorner{X_2}\lrcorner{X_1}\lrcorner\mathop \Omega \limits^5 $ conjugated to V ( $ \mathop \Omega \limits^5 $ is a fixed local volume form on M), the loci of singular behavior of $ V:{M_{\deg }}(V) = \{ p \in M\vert\omega (p) = 0\} $ and $ {M_{{\text{sing}}}}(V) = \{ p \in M\vert\omega \wedge {(d\omega )^2}(p) = 0\} $ are handled. The local classification of this pair of sets is carried out (outside a curve and a discrete set in $ {M_{\deg }}(V)$) up to a smooth diffeomorphism. In the most complicated case, around points of a codimension 3 submanifold of M, $ {M_{{\text{sing}}}}(V)$ turns out to be diffeomorphic to the Cartesian product of $ {\mathbb{R}^2}$ and the Whitney's umbrella in $ {\mathbb{R}^3}$.


References [Enhancements On Off] (What's this?)

  • [AGV] V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of differentiable maps. I, Monographs in Math., vol. 82, Birkhäuser, Boston, Mass., 1985. MR 777682 (86f:58018)
  • [B1] G. R. Belitskii, Normal forms, invariants and local maps, Naukova Dumka, Kiev, 1979. (Russian) MR 537763 (81h:58014)
  • [B2] -, Smooth equivalence of germs of vector fields with a single zero eigenvalue or a pair of purely imaginary eigenvalues, Funktsional Anal. i Prilozhen. 20 (1986), 1-8; English transl. in Functional Anal. Appl. 20 (1986), 253-259. MR 878039 (88f:58125)
  • [F] S. P. Finikov, The Cartan's method of exterior forms in differential geometry, OGIZ, Moscow and Leningrad, 1948. (Russian) MR 0034502 (11:597g)
  • [JP] B. Jakubczyk and F. Przytycki, Singularities of k-tuples of vector fields, Dissertationes Math. (Rozprawy Mat.) 213 (1984). MR 744876 (85k:58013)
  • [Ma] J. Martinet, Sur les singularités des formes différentielles, Ann. Inst. Fourier (Grenoble) 20 (1970), 95-178. MR 0286119 (44:3333)
  • [M1] P. Mormul, Singularities of triples of vector fields on $ {\mathbb{R}^4}$, Bull. Polish Acad. Sci. Math. 31 (1983), 41-49. MR 717124 (85f:58013)
  • [M2] -, Singularities of triples of vector fields on $ {\mathbb{R}^4}$: the focusing stratum, Studia Math. 91 (1988), 241-273. MR 985724 (90c:58028)
  • [M3] -, The involutiveness singularity of 3-distributions in $ T{\mathbb{R}^4}$, preprint.
  • [MR] P. Mormul and R. Roussarie, The geometry of triples of vector fields in $ {\mathbb{R}^4}$, Singularities and Dynamical Systems, (S. N. Pnevmatikos, ed.), North-Holland Math. Studies, vol. 103, Elsevier, Amsterdam, 1985, pp. 89-98. MR 806183 (87m:58124)
  • [W] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63-89. MR 1501735
  • [Z1] M. Ya. Zhitomirskii, Singularities and normal forms of odd-dimensional Pfaffian equations, Funktsional Anal. i Prilozhen. 23 (1989), 70-71; English transl. in Functional Anal. Appl. 23 (1989), 59-61. MR 998435 (90i:58007)
  • [Z2] -, Typical singularities of differential 1-forms and Pfaff equations, Seminar on Supermanifolds, No. 34, (D. Leites, ed.), Reports Dept. Math. Stockholm Univ., 1990, pp. 1-285; slightly revised as Transl. Math. Monographs, vol. 113, Amer. Math. Soc., Providence, RI, 1992. MR 1195792 (94j:58004)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58A17, 58A30, 58C27

Retrieve articles in all journals with MSC: 58A17, 58A30, 58C27


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1150017-9
Keywords: Differential system, class of Pfaffian equation, Morse lemma with parameters
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society