Timeline for A necessary and sufficient condition for a space curve to lie on a ellipsoid
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| Mar 29, 2018 at 2:54 | history | edited | Mohammad Ghomi | CC BY-SA 3.0 |
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| Mar 29, 2018 at 0:11 | comment | added | Wlod AA | @MohammadGhomi, I am sure that the mentioned Borsuk's paper was written before WWII. Hardly any Polish mathematician would write a paper in German during or soon after WWII for obvious reasons, and later because English took over mathematics (with mostly Germans, French, Soviets, and Japanese making an exception for their own languages). | |
| Mar 28, 2018 at 21:57 | history | edited | Mohammad Ghomi | CC BY-SA 3.0 |
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| Mar 28, 2018 at 19:41 | history | edited | Mohammad Ghomi | CC BY-SA 3.0 |
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| Mar 28, 2018 at 18:06 | history | edited | Mohammad Ghomi | CC BY-SA 3.0 |
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| Mar 28, 2018 at 17:00 | history | edited | Mohammad Ghomi | CC BY-SA 3.0 |
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| Mar 28, 2018 at 14:27 | comment | added | Mohammad Ghomi | @Robert, maybe then the integral of the affine torsion would have to vanish? | |
| Mar 28, 2018 at 14:08 | comment | added | Robert Bryant | @MohammadGhomi: Just as in the Euclidean case, there are two affine curvatures of nondegenerate space curves, one of differential order $5$ and one of differential order $6$. (These are the affine analogs of $\kappa$ and $\tau$.) The affine arclength itself is of differential order $3$. | |
| Mar 28, 2018 at 12:34 | comment | added | Mohammad Ghomi | @Robert, Yes $\int \tau=0$ is not affine invariant, since it characterizes only spheres, as opposed to ellipsoids. I think it would be interesting if this has an affine analogue. There are notions such as affine arc length and affine curvature which could relevant. I do not know if someone has yet developed affine torsion. | |
| Mar 28, 2018 at 12:23 | comment | added | Mohammad Ghomi | @Wlod The only reference for this that I know is the paper "Scherrer, W. Eine Kennzeichnung der Kugel. Vierteljschr. Naturforsch. Ges. Zürich 85, (1940)", cited in Millman and Parker. | |
| Mar 28, 2018 at 11:11 | comment | added | Robert Bryant | Note that, except for the criterion $\int\tau\,\mathrm{d}s = 0$, the others are actually affinely invariant conditions, which is appropriate, since the condition of lying on an ellipsoid is affinely invariant. As I pointed out in an earlier comment, it would be more reasonable to seek for conditions in terms of the affine invariants of the curve, and, indeed, the differential equation characterizing $Q$-nondegenerate ellipsoidal curves in terms of affine invariants is much simpler than the one in terms of Euclidean invariants $\kappa$ and $\tau$. | |
| Mar 28, 2018 at 6:00 | comment | added | Wlod AA | I've seen a paper by Karol Borsuk, written in German, in which he proved $\int \tau = 0$ for closed spherical curves. Could you confirm this or was this theorem proven earlier by another geometer? | |
| Mar 28, 2018 at 4:56 | history | edited | Mohammad Ghomi | CC BY-SA 3.0 |
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| Mar 28, 2018 at 3:52 | history | edited | Mohammad Ghomi | CC BY-SA 3.0 |
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| Mar 28, 2018 at 3:39 | history | edited | Mohammad Ghomi | CC BY-SA 3.0 |
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| Mar 28, 2018 at 3:33 | history | edited | Mohammad Ghomi | CC BY-SA 3.0 |
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| Mar 28, 2018 at 3:26 | history | answered | Mohammad Ghomi | CC BY-SA 3.0 |