# Tubular neighborhood theorem

Let $M$ be a submanifold (differential sense) of $\R^n$, of dimension $m$. Then, there exists $\epsilon>0$ such that for any point at distance at most $\epsilon$ from $M$, there is a unique expression of the point as a sum $p + v$ where $p \in M$ and $v$ is a normal at $p$, with $\| v \| < \epsilon$.
If we define $U$ as the open subset of $\R^n$ comprising those points of $\R^n$ at distance less than $\epsilon$ from $M$, then $U$ can be viewed as a concrete realization, in the ambient space $\R^n$, of the normal bundle to $M$ in $\R^n$. In the situations where the normal bundle to $M$ is trivial, we see that this gives a natural diffeomorphism $U \cong M \times \R^{n-m}$.