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I have recently started learning the differential geometry of curves and surfaces.

I am not sure what intrinsic/extrinsic means.

My current understanding is this: an intrinsic quantity on a curve is one that is determined by its shape, so does not change with rotation and translation.

Based on this understanding, the curvature at a point is an intrinsic quantity and the x-coordinate of that point is an extrinsic quantity.

On the other hand, a friend of mine says that curvature is an extrinsic quantity because curves are isometric to straight lines.

What is the standard definition of intrinsic/extrinsic?

MathMan
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    Your friend is correct. For curves, curvature is completely an extrinsic notion. For surfaces, however, Gaussian curvature is not. – Ted Shifrin Aug 14 '23 at 16:47
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    I suggest you take a look at this question, it'll clear up quite a lot: https://math.stackexchange.com/questions/2206328/intrinsic-vs-extrinsic-properties-of-surfaces – Matheus Andrade Aug 14 '23 at 18:26
  • Also, in general, people are much more interested in intrinsic properties than extrinsic ones. There's this thing called the Riemannian curvature tensor which determines intrinsic curvature in general, and it happens that in dimension one it's always zero (basically because antisymmetry and dimension one force it so). As a consequence, any curve is locally isometric to the real line. – Matheus Andrade Aug 14 '23 at 18:29

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