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Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra

  Published:2005-12-01
 Printed: Dec 2005
  • M. Zhitomirskii
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Abstract

In 1999 V. Arnol'd introduced the local contact algebra: studying the problem of classification of singular curves in a contact space, he showed the existence of the ghost of the contact structure (invariants which are not related to the induced structure on the curve). Our main result implies that the only reason for existence of the local contact algebra and the ghost is the difference between the geometric and (defined in this paper) algebraic restriction of a $1$-form to a singular submanifold. We prove that a germ of any subset $N$ of a contact manifold is well defined, up to contactomorphisms, by the algebraic restriction to $N$ of the contact structure. This is a generalization of the Darboux-Givental' theorem for smooth submanifolds of a contact manifold. Studying the difference between the geometric and the algebraic restrictions gives a powerful tool for classification of stratified submanifolds of a contact manifold. This is illustrated by complete solution of three classification problems, including a simple explanation of V.~Arnold's results and further classification results for singular curves in a contact space. We also prove several results on the external geometry of a singular submanifold $N$ in terms of the algebraic restriction of the contact structure to $N$. In particular, the algebraic restriction is zero if and only if $N$ is contained in a smooth Legendrian submanifold of $M$.
Keywords: contact manifold, local contact algebra, relative Darboux theorem, integral curves contact manifold, local contact algebra, relative Darboux theorem, integral curves
MSC Classifications: 53D10, 14B05, 58K50 show english descriptions Contact manifolds, general
Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]
Normal forms
53D10 - Contact manifolds, general
14B05 - Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]
58K50 - Normal forms
 

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