Curved space and curvilinear coordinates

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mertcan
hi, I really wonder what the difference between curvilinear coordinates in a Euclidean space and embedding a curved space into Euclidean space is ? They resemble to each other for me, so Could you explain or spell out the difference? Thanks in advance...

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An embedding is usually of a lower-dimensional manifold. Curvilinear coordinates are used to describe the Euclidean space itself.

You can not introduce Euclidian local coordinates in a curved space

mertcan
I think curvilinear coordinates generally define tangent space, but in curved space also defines normal component besides the tangent space. Am I right? I saw some close definition like this. Is it true?