Local reduction theorems and invariants for singular contact structures

Bronislaw Jakubczyk[1]; Michail Zhitomirskii[2]

  • [1] Polish Academy of Sciences, Institute of Mathematics, Sniadeckich 8, 00-950 Warsaw (Pologne)
  • [2] Technion, Department of Mathematics, 32000 Haifa (Israël)

Annales de l’institut Fourier (2001)

  • Volume: 51, Issue: 1, page 237-295
  • ISSN: 0373-0956

Abstract

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A differential 1-form on a ( 2 k + 1 ) -dimensional manifolds M defines a singular contact structure if the set S of points where the contact condition is not satisfied, S = { p ∈ M : ( ω ∧ ( d ω ) k ( p ) = 0 } , is nowhere dense in M . Then S is a hypersurface with singularities and the restriction of ω to S can be defined. Our first theorem states that in the holomorphic, real-analytic, and smooth categories the germ of Pfaffian equation ( ω ) generated by ω is determined, up to a diffeomorphism, by its restriction to S , if we eliminate certain degenerated singularities of ω (in the holomorphic case they form a set of infinite codimension). We also define other invariants of local singular contact structures: orientations, a line bundle, and a partial connection. We study the problem when these invariants, together with the hypersurface S and the restriction of the Pfaffian equation ( ω ) to S , form a complete set of local invariants. Our results include complete solutions to this problem in dimension 3 and in the case where S has no singularities.

How to cite

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Jakubczyk, Bronislaw, and Zhitomirskii, Michail. "Local reduction theorems and invariants for singular contact structures." Annales de l’institut Fourier 51.1 (2001): 237-295. <http://eudml.org/doc/115911>.

@article{Jakubczyk2001,
abstract = {A differential 1-form on a $(2k+1)$-dimensional manifolds $M$ defines a singular contact structure if the set $S$ of points where the contact condition is not satisfied, $S=\lbrace p\in M : (\omega \wedge (d\omega )^k(p)=0\rbrace $, is nowhere dense in $M$. Then $S$ is a hypersurface with singularities and the restriction of $\omega $ to $S$ can be defined. Our first theorem states that in the holomorphic, real-analytic, and smooth categories the germ of Pfaffian equation $(\omega )$ generated by $\omega $ is determined, up to a diffeomorphism, by its restriction to $S$, if we eliminate certain degenerated singularities of $\omega $ (in the holomorphic case they form a set of infinite codimension). We also define other invariants of local singular contact structures: orientations, a line bundle, and a partial connection. We study the problem when these invariants, together with the hypersurface $S$ and the restriction of the Pfaffian equation $(\omega )$ to $S$, form a complete set of local invariants. Our results include complete solutions to this problem in dimension 3 and in the case where $S$ has no singularities.},
affiliation = {Polish Academy of Sciences, Institute of Mathematics, Sniadeckich 8, 00-950 Warsaw (Pologne); Technion, Department of Mathematics, 32000 Haifa (Israël)},
author = {Jakubczyk, Bronislaw, Zhitomirskii, Michail},
journal = {Annales de l’institut Fourier},
keywords = {contact structure; singularity; pfaffian equation; equivalence; local invariants; reduction theorems; homotopy method},
language = {eng},
number = {1},
pages = {237-295},
publisher = {Association des Annales de l'Institut Fourier},
title = {Local reduction theorems and invariants for singular contact structures},
url = {http://eudml.org/doc/115911},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Jakubczyk, Bronislaw
AU - Zhitomirskii, Michail
TI - Local reduction theorems and invariants for singular contact structures
JO - Annales de l’institut Fourier
PY - 2001
PB - Association des Annales de l'Institut Fourier
VL - 51
IS - 1
SP - 237
EP - 295
AB - A differential 1-form on a $(2k+1)$-dimensional manifolds $M$ defines a singular contact structure if the set $S$ of points where the contact condition is not satisfied, $S=\lbrace p\in M : (\omega \wedge (d\omega )^k(p)=0\rbrace $, is nowhere dense in $M$. Then $S$ is a hypersurface with singularities and the restriction of $\omega $ to $S$ can be defined. Our first theorem states that in the holomorphic, real-analytic, and smooth categories the germ of Pfaffian equation $(\omega )$ generated by $\omega $ is determined, up to a diffeomorphism, by its restriction to $S$, if we eliminate certain degenerated singularities of $\omega $ (in the holomorphic case they form a set of infinite codimension). We also define other invariants of local singular contact structures: orientations, a line bundle, and a partial connection. We study the problem when these invariants, together with the hypersurface $S$ and the restriction of the Pfaffian equation $(\omega )$ to $S$, form a complete set of local invariants. Our results include complete solutions to this problem in dimension 3 and in the case where $S$ has no singularities.
LA - eng
KW - contact structure; singularity; pfaffian equation; equivalence; local invariants; reduction theorems; homotopy method
UR - http://eudml.org/doc/115911
ER -

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