I Parameterized surfaces from coordinates

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

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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.
 
Thanks!
 

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