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THEOREM. Let $f_1,..f_r \in C^\infty (\mathbb R^n$ and $X=\{ x\in\mathbb R^n | f_i(x)=0, 1\le i\le r\}.$ If $$Df_1(x)=(\frac{\partial f_1}{\partial x_1}(x),...,\frac{\partial f_1}{\partial x_n}(x)),\\\vdots\\Df_r(x)=(\frac{\partial f_r}{\partial x_1}(x),...,\frac{\partial f_r}{\partial x_n}(x))$$ are linearly independent for all $x\in X$, then $X$ is an $(n-r)$ dimensional smooth manifold.

Proof:

First, Construct the coordinate charts.

Let $a\in X$, then since the $r\times n$ matrix $\left( \frac{\partial f_i}{\partial x_j}(x) \right)$ has rank $r$ for $x\in X$, and so there exists an $r\times r$ minor of $\left( \frac{\partial f_i}{\partial x_j}(a) \right)$ which is different from $0$. In other words, $\exists ~1\le\alpha_1<...<\alpha_r\le n$ such that $\text{det} \left( \frac{\partial f_i}{\partial x_j}(a)\right)\ne0$. Let $1\le\alpha'_1<...<\alpha'_{n-r}\le n$ be complement to $\alpha_1<...<\alpha_r\le n$.

By the implicit function theorem, there exists open neighborhood $U_{\alpha_1,...,\alpha_r}$ of $a$, an open nbd $V$ of $(a_{\alpha'_{1}},...,a_{\alpha'_{n-r}})$ and a unique map $g:V\to \mathbb R^r$ such that

(1) $g(a_{\alpha'_{1}},...,a_{\alpha'_{n-r}})=(a_{\alpha_1},...,a_{\alpha_r})$

(2) $(x_1,...,x_n) \in X \cap U_{\alpha_1,...,\alpha_r} \Longleftrightarrow (x_{\alpha_1},...,x_{\alpha_r})=g(x_{\alpha'_1},...,x_{\alpha'_{n-r}}).$

Define $\phi_a: X\cap U_{\alpha_1,...,\alpha_r} \to \mathbb R^{n-r},~~(x_1,...,x_n)\to (x_{\alpha'_1},...,x_{\alpha'_{n-r}})$

Then $((x_1,...,x_n) \in X \cap U_{\alpha_1,...,\alpha_r},~\phi_a)$ is a coordinate chart containing $a\in X$.

Second, we need to show that the transition maps are smooth.


I have no idea how to prove the transition maps are smooth. The coordinate chart $\phi_a$ is like a projection map and I think the inverse $\phi^{-1}_a$ should be an inclusion map. Also, another coordinate chart, say $\phi_b$ should also be a projection map and its inverse is an inclusion map as well. Then the transition map $\phi_a \circ \phi_b^{-1}$ will mess up because the projection map and the inclusion map are about the sequences like $\alpha_1,...\alpha_r$ and $\alpha'_1,...\alpha'_{n-r}$. So, I don't know how to write down the transition map explicitly. Also, it seems that in the definition of $\phi_a$, we don't use $f_i$. But $f_i's$ are the only smooth functions that we know. So, I have no idea how to do this proof.

I've just started learning smooth manifolds, so please do not use big theorems. Thank you in advance.

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