Journal of the
Korean Mathematical Society JKMS

Given a system of $s$ independent $1$-forms on a smooth manifold $M$ of dimension $m$, we study the existence of integral manifolds by means of various generalized versions of the Frobenius theorem. In particular, we present necessary and sufficient conditions for there to exist $s'$-parameter $(s' < s)$ family of integral manifolds of dimension $p:=m-s,$ and a necessary and sufficient condition for there to exist integral manifolds of dimension $p'$, $p' \le p$. We also present examples and applications to complex analysis in several variables.

Keywords: Pfaffian system, involutivity, integral manifold, foliation

MSC numbers: 35N10, 58A15, 32F25