Which properties preserve by isometry?

Question

I know that an isometry between two surfaces preserves the 1st fundamental coefficients, geodesic curvature, and Gaussian curvature.

I wonder that how about the curvature and torsion of a curve on the surface, and 2nd fundamental coefficients?

I can’t find any counterexample and prove this question

Give some advice or comments! Thank you!

Once the metric is preserved, every thing depends completely on it is. — Semsem, Nov 18, 2018 at 15:09
The Second Fundamental Form (for example) depends on the embedding of a surface in some higher-dimensional Riemannian space, so it doesn't make sense to ask whether an isometry of surfaces (which makes no reference to embeddings) preserves it. Do you mean perhaps an isometry of some higher-dimensional spaces and the isometry from an embedded surface onto its image? — Travis Willse, Nov 18, 2018 at 16:04
@Travis Just elementary differential geometry level, it means 'in $\mathbb{R}^{3}$ Euclidean space' case. — AnonyMath, Nov 19, 2018 at 6:01
Alright, I've written an answer that assumes you mean an isometry between surfaces, both of which are embedded in $\Bbb R^3$. — Travis Willse, Nov 19, 2018 at 14:40

Travis Willse · Accepted Answer · 2018-11-19 14:40:02Z

1

Hint Consider an isometry from a subset of a plane in $\Bbb R^3$ to a subset of a cylinder in $\Bbb R^3$.

answered Nov 19, 2018 at 14:40

Travis Willse

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$\begingroup$ Consider the A4 paper and drawing a diagonal line on the paper. Then, both of the curvature and torsion of the diagonal line is zero in A4 paper. But, they are non-zero in cylinder in $\mathbb{R}^{3}$ if we bend the A4 paper into cylinder. Right? $\endgroup$
– AnonyMath
Nov 20, 2018 at 6:20
$\begingroup$ It depends on how the paper is bent, but if either of the side pairs are parallel to the axis of the cylinder, for example, then the curve will have nonzero curvature and torsion. (In fact, in this particular example both will be constant.) $\endgroup$
– Travis Willse
Nov 20, 2018 at 6:24
$\begingroup$ you’re right. I just think in my head the line trasforming into cylinderical helix. Thanks for your comments. But, I have one more question. Then, is it true that the isometry preserves 2nd fundamental coefficients? Now, the other properties are clear for me excepts for 2nd fundamental coefficients. $\endgroup$
– AnonyMath
Nov 20, 2018 at 6:40
$\begingroup$ What are the second fundamental forms for the two surfaces in the hint? $\endgroup$
– Travis Willse
Nov 20, 2018 at 6:41
$\begingroup$ Oh, sorry. I have a mistake in calculating of 2nd fundamental coefficients the latter one. Thank you! Now, all things clear! $\endgroup$
– AnonyMath
Nov 20, 2018 at 6:50

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