# Distributions, integral manifold

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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Do you know the definition of an integral manifold for the distribution? (A submanifold $N$ such that the tangent space to $N$ at every point $p\in N$ agrees with $D$ at $p$.) – user31373 Jun 8 '12 at 18:34
A submanifold $(N,\psi)$ of $M$ is an integral manifold of distribution $D$ on $M$ if $d\psi(N_{n})=D(\psi(n))$ for each $n\in N$... but, each point of $M$ there passes an integarl manifold of $D$?? I do not understand! – Manoel Jun 8 '12 at 18:44
OK, so you consider it with the embedding $\psi\colon N\to M$. Then the statement in italics means: for every $p\in M$ there exists $(N,\psi)$ and $n\in N$ such that $\psi(n)=p$. – user31373 Jun 8 '12 at 18:46
Ok I understand ... as a vector $v\in M_{n}$... There is a curve $\gamma$ such that $\gamma(n)=p$ and $\gamma´(n)=v$ Obrigado! – Manoel Jun 8 '12 at 18:58
You're welcome. I'll copy my comment into the answer box, so that the question does not appear unanswered. – user31373 Jun 8 '12 at 19:08

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