This is exercise 1.3.9(b) on Guillemin and Pollack's Differential Topology

I believe I am pretty much done with this problem, but I still do not understand why the last step shows the existence, and what is $g$? Is it $\varphi \circ \varphi^{-1}$?

Assume that $x_1, \dots x_k$ form a local coordinate system on a neighborhood $V$ of $x$ in $X$. Prove that there are smooth functions $g_{k+1}, \dots, g_N$ on an open set $U$ in $\mathbb{R}^k$ such that $V$ may be taken to be the set $$\{(a_1, \dots, a_k, g_{k+1}(a), \dots, g_N(a))\in \mathbb{R}^N: a = (a_1, \dots, a_k) \in U\}. $$

Consider the projection $\varphi: \mathbb{R}^N \rightarrow \mathbb{R}^k$: $$(x_1, \dots, x_N) \mapsto (x_{i_1}, \dots, x_{i_k})$$ Differentiating: $$d \varphi: T_x(X) \mapsto \text{ span}(e_{i_1}, \dots, e_{i_k})$$

Because $\varphi$ is a diffeomorphism, so does its inverse $$\varphi^{-1}: (a_1, \dots, a_k) \mapsto (a_1, \dots, a_k, g_{k+1}(a), \dots, g_N(a)),$$ where $a = (a_1, \dots, a_k) \in U \in \mathbb{R}^k.$

Thank you

  • $\begingroup$ What is $\varphi$? $\endgroup$ – Dan Rust Jul 3 '13 at 16:48
  • $\begingroup$ Remember that Guillemin and Pollack consistently use $\phi$ as a parametrization. What does the first sentence of your problem mean? $\endgroup$ – Ted Shifrin Jul 3 '13 at 16:49
  • $\begingroup$ I am really confused by that as well. I read from somewhere else that $a = (a_1, \dots, a_n)$, then $x_i(a) = a_i.$ $\endgroup$ – WishingFish Jul 3 '13 at 17:15
  • $\begingroup$ Thanks @DanielRust, I added some information :#) $\endgroup$ – WishingFish Jul 3 '13 at 17:16
  • 2
    $\begingroup$ No, Wishing, it is essential that you learn definitions from your reading. Look at the bottom of p. 3 and top of p. 4. You need to explain carefully why $\varphi^{-1}$, in your notation, is of that form. $\endgroup$ – Ted Shifrin Jul 3 '13 at 17:37

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