Defining map

If $S\subset M$ is an embedded submanifold, a smooth map $\phi :M\to N$ is called a defining map if $S$ is a regular level set, i.e. if $S={\phi }^{-1}(\{n\})$ for a regular point $n$.

Local defining map

A map is called locally defining if $\phi :U\to N$ for some subset $U\subset M$ and $S\cap U$ is a regular level set.

Motivation

We call this a defining map, because based on the regular value theorem this map along with the regular value $n\in N$ would define a submanifold.

Relation to tangent space of submanifold

For a submanifold $S\subset M$, if $\Phi :U\to N$ is a local defining map for $S$, then $T_{p}S=\ker d{\Phi }_p:T_{p}M⟶T_{\Phi (p)}N$ for every $p\in S\cap U$.

Proof

todo