A set of coordinates on $B\subseteq\mathbb{R}\times M$, given by a map $\phi:B\rightarrow\mathbb{R}^{n}$, is said to be admissible if the components of $\phi=(\phi_{1},\ldots,\phi_{n})$ satisfy the following properties:

• (i)

  $\phi_{1}$ is the restriction to $B$ of the projection map $\mathbb{R}\times M\rightarrow\mathbb{R}$;

• (ii)

  for $j=2,\ldots,n$ the restrictions of $\phi_{j}$ to $B^{+}$ and $B^{-}$ are locally restrictions of smooth functions on $B$.