Welcome to MathOverflow

Question

0

Let $W\subset \mathbf R^{k}$ be an open set. Are there conditions on $W$ that guarantee the existence of a map $T:(0,1)^k \rightarrow W$ such that: (i) $T$ is surjective, (ii) $T$ is continuously differentiable, and (iii) $T$ has bounded derivative with everywhere nonsingular Jacobian whose determinant is bounded away from zero?

My (naive?) intuition is that $W$ must be bounded, connected and have "smooth" boundary.

Alternatively - is there a characterization of the sets in $\mathbf R^k$ that are diffeomorphic to $\mathbf R^k$?

Thank you in advance!

edited Dec 21 2011 at 6:05

asked Dec 21 2011 at 3:22

Andres
1●1

For the first question, the answer is non-emptiness. $\;$ ($T$ can be chosen to affine.) $\;$ For the second question, that's equivalent to being diffeomorphic to $\mathbf{R}^k$, which seems like a nicer way to ask the question. – Ricky Demer Dec 21 2011 at 3:45

Assuming you also meant to include surjectivity, there are obvious obstructions coming from algebraic topology. – Jack Huizenga Dec 21 2011 at 3:47

2

Ignoring the condition in your title, Ricky Demer answered your question. If you want the map to be a diffeomorphism, then you'd need your space to be contractible among other things. Could you edit your question to make it a little more consistent? – Ryan Budney Dec 21 2011 at 4:32

Hi all: Thank you for your replies! I am sorry for the (very) poor editing. I posted in a rush and forgot two crucial conditions that hopefully make the problem more interesting: 1) The determinant of the Jacobian of $T$ should also be bounded away from zero uniformly on $(0,1)^k$. I believe this is the reason it is not the same as being diffeomorphic to $R^k$? 2) As Jack pointed out, I should have added the map $T$ needs to be surjective. – Andres Dec 21 2011 at 6:00

1

Do you want injectivity? If not, then $W$ could be an annulus in the plane. – Tom Goodwillie Dec 21 2011 at 6:31

show 2 more comments

Sets that are diffeomorphic to $(0,1)^k$

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.