# Ngô Quốc Anh

## December 17, 2009

### R-G: Defining function

Filed under: Riemannian geometry — Ngô Quốc Anh @ 15:58

Let $M$ be a smooth manifold of dimension $n$.

Definition. If $S \subset M$ is an embedded submanifold, a smooth map $\Psi : M \to N$ such that $S$ is a regular level set of $\Psi$ is called a defining map for $S$. In other words,

$\displaystyle S=\Psi^{-1}(c)$

for some point $c \in N$. In particular, if $N=\mathbb R^{n-p}$ (so that $\Psi$ is a real-valued or vector-valued function), it is usually called a defining function.

Example 1. The sphere $\mathbb S^n$ is an embeded submanifold of $\mathbb R^{n+1}$. The sphere is easily seen to be a regular level set of the function $f:\mathbb R^{n+1} \to \mathbb R$ given by $f(x)=|x|^2$ since $df=2 \sum_i x^i dx^i$ vanishes only at the origin.

Definition. More generally, if $U$ is an open subset of $M$ and $\Psi:U \to N$ is a smooth map such that $S \cap U$ is a regular level set of $\Psi$, then $\Psi$ is called a local defining map (or local defining function) for $S$.

Example 2. The smooth map $X:\mathbb R^2 \to \mathbb R^3$ given by

$\displaystyle X(\varphi, \theta)=\big( (2+\cos \varphi) \cos \theta, (2+\cos \varphi) \sin \theta, \sin \varphi\big)$

is an immersion of $\mathbb R^2$ into $\mathbb R^3$ whose image, denoted by $D$, is the doughnut-shaped shape surface obtained by revolving the circle $(y-2)^2+z^2=1$ around the $z$-axis, a point $(x,y,z)$ is in $D$ if and only if it satisfies $(r-2)^2+z^2=1$ where $r=\sqrt{x^2+y^2}$ is the distance from the $z$-axis. Thus $D$ is the zero set of the function $\Psi(x,y,z)=(r-2)^2+z^2-1$ which is smooth on $\mathbb R^3$ minus the $z$-axis. A straightforward computation shows that $lated d\Psi$ does not vanish on $D$, so $\Psi$ is a global defining function for $D$.