Parameterized surfaces from coordinates

[Pencilvester]

For all parameterized (hyper)surfaces that form smooth manifolds of dimension $n-1$ embedded in Euclidean $\mathbb {R}^n$, will there always exist a coordinate system $\partial_{\bar \mu}$ on $\mathbb {R}^n$ that yields the same manifold when the right coordinate (say $\partial_1$) is set to the right constant such that the induced metric on the (sub)manifold is equal to $g_{\bar \mu \bar \nu}$ where any components that have a 1 are dropped?

 

[Orodruin]

Locally, yes. Just take any coordinate system on the surface and use the distance from the surface as your final coordinate. That coordinate system will locally be a coordinate system in the full space.

 

[Pencilvester]

Thanks!