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I am thinking for the simple definition of "foliation" for a manifold. Why foliation is useful in manifold theory?

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    $\begingroup$ Are you asking for the definition? If so, Wikipedia has the answer. $\endgroup$
    – fuglede
    Jun 29, 2013 at 19:43
  • $\begingroup$ I am looking for a simple definition $\endgroup$
    – Matias
    Jun 29, 2013 at 19:48

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Roughly speaking, a codimension $n-q$ foliation $F$ on an $n$-manifold $M$ is partition of $M$ in $q$-manifolds, called leaves, such that locally $M$ is a product $R^{q}\times R^{n-q}$. Foliations are useful because they can give information about the topological structure of the manifold. For example a non-singular foliation on a 2-manifold $M$ implies that $M$ is the torus or the Klein bottle. A special case of a foliation is a non-singular flow, which serves as model for some physical systems.

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    $\begingroup$ I assume the $2$-manifold $M$ is compact? $\endgroup$ Jun 30, 2013 at 20:37
  • $\begingroup$ @JesseMadnick I agree. Otherwise, as it is said on Fantasy Island: "Lookit the plane, boss, lookit the plane!" $\endgroup$
    – coffeemath
    Jul 1, 2013 at 9:54

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