Is extrinsic curvature a property of the manifold or the embedded surface?

Question

Good day, to explain my question further, does changing the way we embed a surface into a manifold change its extrinsic curvature? Just getting exposed to these topics of differential geometry so excuse my ignorance.

Answer 1

One example is to take a paper sheet and draw straight lines on it. You can make a cylinder and a plane with it.

Depending on the direction you draw the lines, they are curves with non-zero curvature in the cylinder, while if you flatten it into a plane, these curves will have zero-curvature in the three dimensional space. In this case both are the same curves, under a transformation preserving distances since you do not stretch the paper.

So any extrinsic measure of curvature can depend on how we embed it on the manifold.