Submanifolds again.

If $M$ is a manifold then we can define a submanifold of $M$ by using the principal property of submanifolds in $\mathbb{R}^n$.

Definition 4.4 (Submanifolds.)   We say that a subset $N \subset M$ is a submanifold of dimension $d$ of a manifold $M$ of dimension $m$ if for every $x \in N$ we can find a co-ordinate chart $(U, \psi)$ for $M$ with $x \in N$ and such that

$$U \cap N = \{ y \in N \colon \mid \psi^{d+1}(y)= \dots = \psi^n(y) = 0\}.$$

Just as before we can define co-ordinates $(U\cap N, \bar\psi)$ on $N$ by letting

$$\bar\psi^i(y) = \psi^i(y)$$

for each $i = 1, \dots, d$. Similarly we have

Proposition 4.4   The set consisting of all the charts $(U\cap N, \bar\psi)$ constructed in this manner is an atlas. Moreover it makes $N$ a manifold in such a way that the inclusion map $\iota_N \colon N \to M$ defined by $\iota_N(n) = n$ is smooth.

Because the condition for being a submanifold is local we can use the inverse function theorem as in Proposition 4.2 to prove

Proposition 4.5   Let $f \colon M \to N$ be a smooth map between manifolds of dimension $m$ and $n$ respectively. Let $n \in N$ and $Z = f^{-1}(n)$. Then if $T_zf$ is onto for all $z \in M$ the set $Z$ is submanifold of $M$. Moreover the image of $\iota_Z$ in $T_zM$ is precisely the kernel of $T_zF$.