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Transactions of the American Mathematical Society
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
MathSciNet review: 1150017
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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}$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1150017-9
PII: S 0002-9947(1994)1150017-9
Keywords: Differential system, class of Pfaffian equation, Morse lemma with parameters
Article copyright: © Copyright 1994 American Mathematical Society