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I am looking for an example showingthat a function $f$ which is $C^\infty$ on a submanifold $N$ of $M$, but it cannot be written as the restriction of a $C^\infty$-function on $M$.

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    $\begingroup$ Assuming $N$ is meant to be closed in $M$ (you don't clarify your definition of "submanifold"), there are no such examples (i.e., every smooth function on $N$ lifts to one on $M$), due to constructions using partitions of unity and the openness of $M-N$ in $M$ to patch local lifts into a global one. Of course, if you allow open sub manifolds then there are trivial examples such as $1/x$ on $N = \mathbf{R}^{\times}$ inside $M = \mathbf{R}$. $\endgroup$
    – user30379
    Jan 5, 2013 at 15:39
  • $\begingroup$ Yes. BE a submanifold, I mean an open submanifold. Thanks. $\endgroup$
    – hamid
    Jan 5, 2013 at 17:35
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    $\begingroup$ The absolute value of $x$ on the nonzero reals. $\endgroup$ Jan 6, 2013 at 0:09
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    $\begingroup$ Or $1/x$ on the nonzero reals. $\endgroup$ Jan 6, 2013 at 17:42

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You can take $M = \{ z \in \mathbb{C} \colon |z|=1\}$, $N = M \setminus \{1\}$ and $f(e^{i\varphi}) = \varphi$.

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