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The question is to find all possible curvature for any smooth curve in a donut. I parametrize the donut as $$((3+\sin v)\sin u, (3+\sin v)\cos u, \cos v)$$

My thought is, since if we keep shrinking a circle, curvature goes to infinity, we only need to find the minimum curvature of any normal section. But I have trouble finding that...

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  • $\begingroup$ All curvatures? There are many curvatures : Normal curvature, principal normal curvatures, geodesic curvature, Gaussian curvature, Mean curvature, Integral curvature, geodesic torsion.which are some common curvatures. $\endgroup$ – Narasimham 2 days ago
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If instead of unit normal curvature we have a different normal curvature of the donut ( produced by tightening a string around the tube meridional section for instance ) then its parameterization would be:

$$ [ (3+ \frac{\sin v}{kn})\sin u, (3+ \frac{\sin v}{kn} )\cos u, \frac{\cos v}{kn}]$$

Yes, when $ kn \to \infty $ we are left with a thin circle around the middle of torus.

It is then no more a 2D surface embedded in 3-space, but just a line in 3-space.

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