Foliations associated with Pfaffian systems
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Given a system of smooth 1-forms θ = (θ^1,…,θ^s) on a smooth manifold Mm, we give a necessary and sufficient condition for M to be foliated by integral manifolds of dimension n, n ≤ p := m −s, and construct an integrable supersystem (θ,η) by finding additional 1-forms η = (η^1, . . . , η^(p−n)). We also give a necessary and sufficient condition for M to be foliated by reduced submanifolds of dimension n, n ¸ p, and construct an integrable subsystem (dρ^1, . . . , dρ^(m−n)) by finding a system of first integrals ρ = (ρ^1, . . . , dρ^(m−n)). The special case n = p is the Frobenius theorem on involutivity.
Abstract
Given a system of smooth 1-forms θ = (θ^1,…,θ^s) on a smooth manifold Mm, we give a necessary and sufficient condition for M to be foliated by integral manifolds of dimension n, n ≤ p := m −s, and construct an integrable supersystem (θ,η) by finding additional 1-forms η = (η^1, . . . , η^(p−n)). We also give a necessary and sufficient condition for M to be foliated by reduced submanifolds of dimension n, n ¸ p, and construct an integrable subsystem (dρ^1, . . . , dρ^(m−n)) by finding a system of first integrals ρ = (ρ^1, . . . , dρ^(m−n)). The special case n = p is the Frobenius theorem on involutivity.
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