Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2009; 46(5): 931-940

Printed September 1, 2009

https://doi.org/10.4134/BKMS.2009.46.5.931

Copyright © The Korean Mathematical Society.

Foliations associated with Pfaffian systems

Chong-Kyu Han

Seoul National University

Abstract

Given a system of smooth $1$-forms $\theta=(\theta^1,\ldots,\theta^s) $ on a smooth manifold $M^m$, we give a necessary and sufficient condition for $M$ to be foliated by integral manifolds of dimension $n$, $ n\le p:= m-s, $ and construct an integrable supersystem $(\theta,\eta)$ by finding additional $1$-forms $\eta=(\eta^1, \ldots,\eta^{p-n}).$ We also give a necessary and sufficient condition for $M$ to be foliated by reduced submanifolds of dimension $n$, $n\ge p,$ and construct an integrable subsystem $(d\rho^1,\ldots,d\rho^{m-n})$ by finding a system of first integrals $\rho=(\rho^1,\ldots,\rho^{m-n}).$ The special case $n=p$ is the Frobenius theorem on involutivity.

Keywords: Pfaffian system, integral manifolds, reduced manifolds, foliation, Frobenius integrability

MSC numbers: 35N10, 58A15, 32F25