Prove that there exists a diffeomorphism from an open neighborhood of $Z$ in $N(Z;Y)$ (normal bundle of $Z$ in $Y$) onto an open neighborhood of $Z$ in $Y$.
$\epsilon$ neighborhood theorem: For a compact boundaryless manifold $Y$ in $R^M$ and $\epsilon >0$, let $Y^\epsilon$ be the open set of points in $R^k$ with distance less than $\epsilon$ from $Y$. If $\epsilon$ is sufficiently small then each point $w\in Y^\epsilon$ possesses a unique closest point in $Y$, denoted $\pi(w)$. Moreover the map $\pi : Y^\epsilon \to Y$ is a submersion.
Proof: Let $Y$ be a compact boundaryless manifold in $R^M$. For any $ϵ >0$ , let $Y^ϵ$ be the open set of point in $R^M$ with distance less than $ϵ$ from $Y$. Let $π:Y^ϵ→Y$ where $Y^ϵ=\{w∈R^M:|w-y|<ϵ(y) for some y∈Y\}$. By the $ϵ$-neighborhood theorem, $π$ is submersion . Let $Z$ be any boundaryless submanifold of $Y$. Let $h:N(Z,Y)→R^M$ be defined by $h(z,v)=z+v$. Then $h$ is regular at every point of $Z×{0}$ in $N(Z,Y)$ because at $(z,0)$ there are 2 natural complementrary manifold of $N(Z,Y)$ pass through ie $Z×{0}$ and ${z}×N_z (Z)$. The derivative of $h$ at $(z,0)$ map the tangent space of $Z×{0}$ at $(z,0)$ onto $T_z (Z)$ and map the tangent space of ${z}×N_z (Z)$ at $(z,0)$ onto $N_z (Z)$. There for it maps onto $T_z (Z)+N_z (Z)=R^M$
now I'm stuck