Reference for working with the implicit function theorem

Question

I just had a student come to my office hours and ask me a ton of questions, the answer to all of which was "that's a slight variant to the implicit function theorem, which is proved by formal manipulation from the implicit function theorem". For example:

$\def\RR{\mathbb{R}}$ Let $f : \RR^n \to \RR$ be a smooth function and suppose that $\partial f/ \partial x_n \neq 0$. Then, locally, $(x_1, x_2, \ldots, x_{n-1}, f)$ are coordinates on $\RR^n$.

Let $f : \RR^n \to \RR$ be a smooth function and suppose that $\partial f/ \partial x_n \neq 0$. Then, locally, $(x_1, x_2, \ldots, x_{n-1})$ are coordinates on $\{ f=0 \}$.

Let $U$ be a small open set in $\RR^d$ and let $(f_1, \ldots, f_d): U \to \RR^n$ parameterize a patch on a manifold $M$ in $\RR^n$. Suppose that $\det (\partial f_i/\partial x_j)_{1 \leq i,j \leq d} \neq 0$. Then $x_1$, ..., $x_d$ are local coordinates on $M$.

Let $g_1$, ..., $g_{n-d}$ be smooth functions $\RR^n \to \RR$. Let $M= \{g_1=g_2=\ldots=g_{n-d} = 0 \}$. Suppose that $\det (\partial f_i/\partial x_j)_{1 \leq i,j \leq n-d} \neq 0$. Then $M$ is a smooth manifold of dimension $d$ and $x_{n-d+1}$, ..., $x_n$ are local coordinates near $0$.

Does anyone know a book which works through these sort of variants systematically?

I should mention that I actually need these facts for holomorphic functions. But I have a good reference for the holomorphic implicit function theorem: Gunning and Rossi, Chapter 1. The problem is that I want a reference which goes slowly through these variants, rather than assuming they are obvious corollaries.

Answer 1

Shameless plug: In my thesis, I introduced the following notion of an (abstract) normal form. It consists of a tuple $(X, Y, \hat{f}, f_s)$ where:

• $X$ and $Y$ finite-dimensional vector spaces with compositions $X = Ker \oplus Coim$ and $Y = Coker \oplus Im$,
• $\hat{f}: Coim \to Im$ is a linear isomorphism,
• $f_s: X \to Coker$ is a smooth map with vanishing derivative at $0$ which satisfies $f_s(0, x) = 0$ for all $x \in Coim$.

The inverse function theorem can then be used to show that every smooth map $f: M \to N$ can be brought into such a normal form, i.e. there exist charts such that $f$ locally coincides with $\hat{f} + f_s$. The abstract spaces $Ker$, $Coimg$, $Coker$ and $Im$ in the normal form are then of course identified with the corresponding kernel, coimage, ... of the derivative of $f$. This normal form theorem has the immersion, the submersion and the constant rank theorem as direct corollaries, and it also serves as the natural "parent" for all your statements.

For example consider the second statement.

Let $f : \mathbb R^n \to \mathbb R$ be a smooth function and suppose that $\partial f/ \partial x_n \neq 0$. Then, locally, $(x_1, x_2, \ldots, x_{n-1})$ are coordinates on $\{ f=0 \}$.

Proof: Since $\partial f/ \partial x_n \neq 0$, we have $Ker = span (x_1, \dots, x_{n_1})$, $Coimg = span (x_n)$, $Img = \mathbb R$ and trivial $Coker$. Moreover, $f$ is locally given by a linear isomorphism $\hat{f}: Coimg \to Img$ so that $\{f = 0\}$ is identified with $Ker = span (x_1, \dots, x_{n_1})$.

Answer 2

Well, not to all of them, but nevertheless a nice approach: in the differential topology book by Bröcker and Jänich, they discuss various applications of the implicit function theorem and the theorem of constant rank maps, using them to build coordinate systems etc. Maybe this is worth a look. I only have the german edition (there it is in Chap 5) but I think there is an english version around. They formulate it for the real/smooth setting, though :( But the ideas are the same of course.

Answer 3

I have a good reference, which is even available online, but something tells me you will cry : the wikipedia page on the Théorème des fonctions implicites is more complete than its english counterpart!

Notice that they're refencing a Lang book... so perhaps that will do better.

Answer 4

Perhaps The Implicit Function Theorem by Krantz and Parks. Not a textbook, but quite interesting.