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Let $M$ be a differentiable manifold of dimension $n + k$, let $\Delta$ be an $n$-dimensional integrable distribution (à la Frobenius), let $N$ be an $n$-dimensional connected integral manifold of $\Delta$, and assume that $N$ is closed in the ambient manifold $M$. It seems that $N$ should be an embedded submanifold of $M$ (not only immersed), how do you prove this?

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1 Answer 1

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Choose a point $p$ in $N$, and choose coordinates for $M$ near $p$ such that $N$ appears as $\mathbb R^n\times S$ for some $S\subset \mathbb R^k$. The set $S$ is closed, countable, and nonempty, so by the Baire category theorem it must have an isolated point.

This implies that some nonempty open subset of $N$ is embedded in $M$. Since the group of diffeomorphisms of $M$ preserving the foliation and the leaf $N$ acts transitively on $N$, $N$ is embedded.

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