Timeline for Conditions for a curve to belong to a hyper-surface in $\mathbb{R}^n$
Current License: CC BY-SA 3.0
4 events
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| Mar 19, 2014 at 17:22 | comment | added | Robert Bryant | Yes, but, again, in this case, the invariance of the hypersphere under a rotation group guarantees that there are degeneracies of the $k$-jet mappings involved, so, of course it simplifies in that case, just as is does in the case of a regular sphere in $3$-space. The case of a general hypersurface is much more complicated than for a hypersphere. | |
| Mar 19, 2014 at 16:59 | comment | added | Ayan | Thank you very much for the answer. There are specific results in $\mathbb{R}^3$ for a curve, with nonzero torsion and curvature, to lie on a hyper-sphere. It can be easily generalized to $\mathbb{R}^6$ by using generalized Serret-Frenet formulas. These conditions are simply statements about the complete order of contact between the given curve and a hyper-sphere. | |
| Mar 19, 2014 at 16:47 | vote | accept | Ayan | ||
| Mar 19, 2014 at 15:23 | history | answered | Robert Bryant | CC BY-SA 3.0 |