Local defining functions on real hypersurfaces.

Question

I'm currently reading Real Submanifolds in Complex Space and their Mappings by Baouedi, Ebenfelt and Rothschild.

I'm currently stumped by what the author's claim to be an easy check (really blowing my confidence): on page 4, he says that if $\rho$ and $\rho'$ are two defining functions for the real hypersurface $M$ near $p_0$, then there is a non vanishing real-valued smooth function $a$ defined in a neighbourhood of $p_0$ such that $\rho = a \rho'$ near $p_0$.

I don't even know where to start to verify this seemingly easy fact.

Any hint is much appreciated!

Answer 1

Here is a general theorem which in particular answers your question:

The differentiable Nullstellensatz:
Let $M$ be a differentiable manifold and $S\subset M $ a submanifold given by $S=f^{-1}(0)$ for some differentiable $f\in C^\infty(M)$ with $df(s)\neq0$ for all $s\in S$.
Then any $G\in C^\infty(M)$ can be written $G=fg$ for some $g\in C^\infty(M).$
Proof
Of course we define $g:=\frac Gf$ outside $S$ and the problem is to show that $g$ extends to all of $M$ in a smooth way.
This is a local problem, so we may assume (by the constant rank theorem) that $M=\mathbb R^n$, $f(x_1,\cdots,x_n)=x_1$ and $S=\{0\}\times \mathbb R^{n-1}$.
For smoothness near, say, the origin $(0,\cdots,0)$ we just write down the formula $$G(x_1,x_2,\cdots,x_n)= G(0,x_2,\cdots,x_n)+x_1 \int _0^1 \frac {\partial {G}}{\partial x_1}(tx_1,x_2,\cdots,x_n) dt $$ Since $G(0,x_2,\cdots,x_n)=0$ by hypothesis, we see that $\frac Gf=\frac {G}{x_1}$ a priori only defined for $x_1\neq 0$ is extended to a smooth function $g\in \mathbb C^\infty (\mathbb R^n)$ by the everywhere defined function $$g(x_1,\cdots,x_n)=\int _0^1 \frac {\partial {G}}{\partial x_1}(tx_1,x_2,\cdots,x_n) dt $$

Remark
This is the analogue, sometimes attributed to Hadamard, of the Nullstellensatz, a basic theorem in algebraic geometry.
(The condition $df(x)\neq 0$ is a substitute for the requirement that the ideal $\langle f\rangle$ be radical).

A personal opinion
I find it regrettable that this basic result is not emphasized more in books and courses on differential manifolds.

How simple it all is!
The only mathematical content of the above is the elementary one variable calculus formula $$G(x)=G(0)+x\int _0^1 G'(tx) dt$$ with some harmless parameters $x_2,\cdots,x_n$ thrown in.

Answer 2

The main issue here is that if $\tau$ is a defining function for a hypersurface $M$ and $f$ is any smooth function which vanishes on $M$, then $f=\tilde f\tau$ for some smooth function $\tilde f$ locally around $M$. You have to apply this to $\tau=\rho'$ and $f=\rho$, to get $\rho=a\rho'$. Then computing the derivative $d\rho=\rho'da+ad\rho'$, the first summand vanishes along $M$, so $d\rho$ can only be non-vanishing along $M$ if $a$ is non-vanishing along $M$ (and hence on a neighborhood of $M$).

To prove the claim that $f=\tilde f\tau$ you use that you can use $\tau$ as a local coordinate for local identifications of a neighborhood of $M$ with $M\times (-\epsilon,\epsilon)$. To do this, take a chart in $M$, i.e. an open subset $U$ in $M$ and a diffemorphism $u$ from $U$ to an open subset $u(U)\subset \mathbb R^{2N-1}$. Extend the components of $U$ arbitrarily to some neighborhood $V$ of $U$ in $\mathbb C^N$ (denoting them by the same letter) and consider $(u,\rho):V\to\mathbb R^{2N}$. By definition of a defining function, this has invertible differential in each point $x\in M\cap V$ and hence defines a local chart on a neighborhood of $x$ in $\mathbb C^N$, whose last coodinate is $\tau$.

This reduces the question to the statement that a smooth function $f:(-\epsilon,\epsilon)\to\mathbb R$ such that $f(0)=0$ can be written as $f(t)=t\tilde f(t)$ for a smooth function $\tilde f$, which is a simple consequence of Taylor's theorem.