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I do not understand the following: "Let D be a smooth distribution on M such That through each point of M there passes an integral manifold of D". I understand the distribution of M, I do not understand: through each point of M there passes an integral manifold of D.

thank you.

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    You're welcome. I'll copy my comment into the answer box, so that the question does not appear unanswered.2012-06-08

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The statement in italics means: for every $p\in M$ there exists an integral submanifold $(N,\psi)$ and $n\in N$ such that $\psi(n)=p$.