2
$\begingroup$

I have a question to the end of my proof for the problem 1.3.10 on Guillemin and Pollack's Differential Topology:

Generalizaition of the Inverse Function Theorem: Let $f: X \rightarrow Y$ be a smooth map that is injective on a compact sumbanifold $Z$ of $X$. Suppose that for all $x \in Z,$ $$df_x: T_x(X) \rightarrow T_{f(x)}(Y)$$ is an isomorphism. Then $f$ maps $Z$ diffeomorphically onto f(Z). Why? Prove that $f$ maps an open neighborhood of $Z$ in $X$ diffeomorphically onto an open neighborhood of $f(Z)$ in $Y$.

Suppose we have $$U_i = \cup_{z\in Z} B_i(Z),$$ where each $B_i$ is an open neighborhood centered at $z$, with $B_i(z) \rightarrow z$ as $i \rightarrow \infty$. Placing a metric on $X$ induced by $\mathbb{R}^N$ and take $B_i(z) = B(z, 1/i)$ for each $i \in \mathbb{N}$. Clearly $Z \subset U_i$ and $U_i$ is open. If $f$ is not one-to-one on some neighborhood of $Z$, we can find sequences of points $\{a_i\}, \{b_i\} \subset U_i$ such that $a_i \neq b_i$ and $f(a_i) = f(b_i).$ Since $Z$ is a compact manifold, the sequence $\{a_i\}, \{b_i\}$ is bounded, there exist convergent subsequences $a_i \to a, b_i \to b$ in $Z$.

Then I got lost here: How can I show that $a,b \in Z$?

$\endgroup$
  • 3
    $\begingroup$ are you aware of the fact that compact submanifolds are also sequentially compact? $\endgroup$ – Eric O. Korman Jul 4 '13 at 2:05
  • 1
    $\begingroup$ Yes @EricO.Korman, but I got lost considering all $a_i$s and $b_i$s just happen to be chosen in $U_i$ but outside of $Z$? Since only sequence in $Z$ converge $Z$, this case fails to converge in $Z$. $\endgroup$ – WishingFish Jul 4 '13 at 2:50
3
$\begingroup$

I guess some wording elsewhere confused me. Anyways, it just by the definition of compactness, $Z$ contains all of its limit points. Therefore, for any converging sequences $a_i \to a,b_i \to b$ in $Z$, $a,b \in Z.$

$\endgroup$

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.