2
$\begingroup$

If $F:\overline{M} \to \overline{M'}$ is a diffeomorphism between two open bounded domains $M$ and $M'$ in $\mathbb{R}^n$, what conditions do I need on $F$ or $M$ or $M'$ to make sure that $F(\partial M) = \partial M'$?

Or is this simply something that I must specify? Thanks

$\endgroup$
7
  • $\begingroup$ open domaine don't have boundries. $\endgroup$
    – Wintermute
    Jan 1, 2014 at 15:35
  • 1
    $\begingroup$ ?? $(0,1)$ has boundary $\{0,1\}.$ $\endgroup$ Jan 1, 2014 at 15:36
  • $\begingroup$ your diffeomorphism are not defined on the boundary. Do you want to find an extension with such a property? $\endgroup$ Jan 1, 2014 at 15:40
  • $\begingroup$ @EmanuelePaolini Oops. I meant it's defined on closure of $M$. Let me edit. $\endgroup$ Jan 1, 2014 at 15:42
  • $\begingroup$ I think that a diffeomorphism between manifolds with boundary has the requested property by definition. $\endgroup$ Jan 1, 2014 at 15:46

1 Answer 1

2
$\begingroup$

Every homeomorphism between two bounded open sets $M$ and $M'$ in $\mathbb R^n$ induces a boundary correspondence in the following sense:

  • $x\in \partial M$ corresponds to the cluster set $$\{y\in \mathbb R^n: \exists (x_n) \to x \text{ such that } F(x_n)\to y\}$$

The following properties hold:

  1. Each cluster set is a nonempty closed subset of $\partial M'$
  2. The union of cluster sets over $x\in\partial M$ is equal to $\partial M'$.

Proofs:

  1. "Nonempty and closed" are easy to see. Suppose $y$ is an element of a cluster set and $y\notin \partial M'$. Then $y\in M'$. Let $U$ be a neighborhood of $y$ such that $\overline{U}\subset M'$. The set $F^{-1}(\overline{U})$ is a compact subset of $M$. Any sequence $x_n$ in $M$ that approaches the boundary of $M$ will eventually leave $F^{-1}(\overline{U})$; thus, $F(x_n)$ will eventually be outside of $U$. Contradiction.

  2. Take $y\in\partial M'$, pick a sequence $y_n\to y$ in $M'$, then choose a convergent subsequence of $F^{-1}(y_n)$.

So far, no assumptions on $F$ were needed besides it being a homeomorphism. If you want to have boundary correspondence as a map, point-goes-to-a-point, then add the assumption that for every $x\in\partial M$ the limit $\lim_{z\to x}F(x)$ exists. This limit, which can also be denoted $F(x)$, satisfies $F(\partial M ) = \partial M'$ by the above.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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