MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $M$.

share|cite|improve this question
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 $M = \mathbf{R}$. – user30379 Jan 5 '13 at 15:39
Yes. BE a submanifold, I mean an open submanifold. Thanks. – hamid Jan 5 '13 at 17:35
The absolute value of $x$ on the nonzero reals. – Lunasaurus Rex Jan 6 '13 at 0:09
Or $1/x$ on the nonzero reals. – Gerald Edgar Jan 6 '13 at 17:42
up vote 3 down vote accepted

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.