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- Mar 10, 2012

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I need your help to confirm that I have got the proof right of a very important theorem.

Theorem 2-13 in Spivak's Calculus on Manifolds.

Theorem 2-13 in Spivak's Calculus on Manifolds.

Let $p\leq n$ and $f:\mathbb R^n\to\mathbb R^p$ be a continuously differentiable function in an open set $O$ of $\mathbb R^n$.

Let $a$ be a point in $O$ such that $f(a)=0$ and assume that the $p\times n$ matrix $M$ with the $i,j$-th entry $[M]_{i,j}=D_jf_i(a)$ has rank $p$.

Then there exists an open set $A$ of $\mathbb R^n$ which contains $a$, and a diffeomorphism $h:A\to\mathbb R^n$ such that $f\circ h(x_1,\ldots,x_n)=(x_{p-n+1},\ldots,x_n)$.

In Spivak's book, it says that Theorem 5.1 is immediate using the above theorem. I have tried to prove a slight variation of Theorem 5.1 below.

**Notation:**

Let $x\in \mathbb R^n$. We write $x_k^+$ as a shorthand for $(x_{k+1},\ldots,x_n)$ and $x_k^-$ as a shorthand for $(x_1,\ldots,x_{k-1})$.

**To Prove:**

Let $p\leq n$ and $f:\mathbb R^n\to\mathbb R^p$ be a continuously differentiable function such that $Df(x)$ has rank $p$ whenever $f(x)=0$.

Then $f^{-1}(0)$ is a $(n-p)$-dimensional manifold in $\mathbb R^n$.

**Proposed Proof:**

Let $a=(a_1,\ldots,a_n)$ be in $f^{-1}(0)$.

Then $f(a)=0$ and thus by Spivak's Theorem 2.13 there exists an open set $A$ of $\mathbb R^n$ which contains $a$, and a diffeomorphism $h:A\to\mathbb R^n$ such that $f\circ h(x)=x_{n-p}^+$.

Write $M=\{x\in A:x_{n-p}^+=0\}$.

We now show that $h(M)=f^{-1}(0)$.

*Claim 1:*$h(M)\subseteq f^{-1}(0)$.

*Proof:*

Let $y\in M$.

Then $f\circ h(y)=y_{n-p}^+$ and since $y_{n-p}^+=0$, we have $f(h(y))=0$.

Therefore $f(h(M))=\{0\}$.

This gives $h(M)\subseteq f^{-1}(0)$ and the claim is settled.

*Claim 2:*$f^{-1}(0)\subseteq h(M)$.

*Proof:*

Let $x\in f^{-1}(0)$.

Since $h$ is a diffeomorphism, it is a bijection and thus there is $y\in A$ such that $h(y)=x$.

Thus $f(h(y))=f(x)=0$.

This means $f\circ h(y)=0$, that is $y_{n-p}^+=0$, meaning $y\in M$.

Hence $x\in h(M)$ and since $x$ was arbitrarily chosen in $f^{-1}(0)$, we conclude that $f^{-1}(0)\subseteq h(M)$ and the claim is settled.

From the above two claims we have shown that $f^{-1}(0)=h(M)$.

Note that $M=A\cap \{x\in \mathbb R^n:x_{n-p}^+=0\}$.

Since $A$ is an open set in $\mathbb R^n$ and $\{x\in \mathbb R^n:x_{n-p}^+=0 \}$ is a $(n-p)$-dimensional manifold in $\mathbb R^n$, we infer that $M$ is a $(n-p)$-dimensional manifold in $\mathbb R^n$.

Now since $h$ was a diffeomorphism, and since diffeomorphisms take manifolds to manifolds and preserve the dimension, we know that $h(M)$ is a $(n-p)$-dimensional manifold in $\mathbb R^n$.

Having already shown that $h(M)=f^{-1}(0)$, our lemma is proved.

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Can anybody please check the proof and confirm that it's correct or else point point out the errors?

Thanks in advance for taking the time out.