Let $M,N$ be manifolds and $f:N\to M$ an embedding. For every $p\in N$, $df_p:T_pN\to T_pM$ shows that $T_pN\subset T_pM$. For every $p\in N$, define the normal space of $N$ in $M$ at $p$ as $$ (\eta_{N|M})_p:=\frac{T_pM}{T_pN}. $$ Gluing together all these spaces, we obtain $$ \eta_{N|M}:=\frac{TM|_N}{TN} $$ the normal bundle of $N$ in $M$.
The Tubular Neighbourhood Theorem holds.
Let $S$,$M$ be manifolds without boundary, $i:S\to M$ an embedding. Then there exists
- $T$, neighborhood of $i(S)$ in $M$
- $v:\eta_{S|M}\to T$ such that $i=v\circ s_0$, where $s_0:S\to \eta_{S|M}$ is the zero section.
Now, let $M$ be a manifold with our boundary, $f:M\to\mathbb{R}$ be a smooth function, $y\in Reg(f)$ and $N$ be a connected component of $f^{-1}(y)$. I have to prove that $\eta_{N|M}$ is trivial.
I can't understand how to proceed. First of all because the definition of normal bundle is not clear to me at all.
Showing that the normal bundle is trivial is the same thing as demonstrating a nonvanishing global section of $\eta$ (since it's one-dimensional). At each point $n \in N$ let $v \in (TM/TN)_n$ be the unique vector with $df(v) = 1 \in T_y \Bbb R$. (Why is there a unique such vector?) This is a continuous section of the normal bundle, as you can check in local coordinates (pick a slice chart for the submersion). Because this section cannot vanish anywhere (it maps to something nonzero!), this is the desired trivializing section.