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I am strugling one of the problems of "Differential Topology" by Guillemin:
Suppose that $Z$ is an $l$-dimensional submanifold of $X$ and that $z\in Z$. Show that there exsists a local coordinate system $\left \{ x_{1},...,x_{k} \right \}$ defined in a neighbourhood $U$ of $z$ such that $Z \cap U$ is defined by the equations $x_{l+1}=0,...,x_{k}=0$.
I assume that the solution should be based on the Local Immersion theorem, which states that "If $f:X\rightarrow Y $ is an immersion at $x$, then there exist a local coordinates around $x$ and $y=f(x)$ such that $f(x_{1},...,x_{k})=(x_{1},...,x_{k},0,...,0)$".
I would really appreciate any pointers on how to attack this problem.

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Because $Z$ is a submanifold of $X$ you know that the inclusion $i:Z\rightarrow X$ is a immersion. Now you can use the result that you have stated. –  Tomás Dec 27 '12 at 13:50
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Thanks, that was helpful. –  Tomas Dec 27 '12 at 14:01

1 Answer 1

up vote 2 down vote accepted

I found answers from Henry T. Horton at the question Why the matrix of $dG_0$ is $I_l$. and Augument, and injectivity. very helpful for solving this question.

To repeat your question, which is found in Guillemin & Pallock's Differential Topology on Page 18, problem 2:

Suppose that $Z$ is an $l$-dimensional submanifold of $X$ and that $z \in Z$. Show that there exists a local coordinate system {$x_1, \dots, x_k$} defined in a neighborhood $U$ of $z$ in $X$ such that $Z \cap U$ is defined by the equations $x_{l+1}=0, \dots, x_k=0$.

Here is my attempt Consider the following diagram:

$$\begin{array} AX & \stackrel{i}{\longrightarrow} & Z\\ \uparrow{\varphi} & & \uparrow{\psi} \\ C & \stackrel{(x_1, \dots, x_l) \mapsto (x_1, \dots, x_l, 0, \dots, 0)}{\longrightarrow} & C^\prime \end{array} $$

Since $$i(x_1, \dots, x_l) = (x_1, \dots, x_l,0,\dots, 0) \Rightarrow di_x(x_1, \dots, x_l) = (I_x, 0,\dots, 0).$$Clearly, $di_x$ is injective, thus the inclusion map $i: X \rightarrow Z$ is an immersion.

Then we choose parametrization $\varphi: C \rightarrow X$ around $z$, and $\psi: C^\prime \rightarrow Z$ around $i(z)$. The map $C \rightarrow C^\prime$ sends $(x_1, \dots, x_l) \mapsto (x_1, \dots, x_l, 0, \dots, 0)$.

The points of $X$ in a neighborhood $\varphi(C)$ around $z$ are those in $Z$ such that $x_{l+1} = \cdots x_k=0$, and this concludes the proof.

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