Foliations associated with Pfaffian systems
Bull. Korean Math. Soc. 2009 Vol. 46, No. 5, 931-940
https://doi.org/10.4134/BKMS.2009.46.5.931
Printed September 1, 2009
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
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