Timeline for An almost complex structure on $\Bbb S^n$ induces a cross product on $\Bbb R^{n+1}$
Current License: CC BY-SA 4.0
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| Dec 20, 2023 at 7:54 | comment | added | Kevin Casto | (oops, and scale by $\|a\|$) | |
| Dec 20, 2023 at 7:24 | comment | added | Kevin Casto | I guess the construction would go something like: given the first vector $a$, we can view the orthogonal complement to $a$ as $T_{\hat a} S^n$. Then project $b$ onto this subspace and use the almost complex structure to rotate it 90°. Does that work? | |
| Dec 20, 2023 at 7:20 | comment | added | Ryan Budney | It seems the more natural implication goes the other way. If you have a cross product you get the almost complex structure $J$ as rotation by $\pi/2$ using the basis $v, p \times v$ in $T_p S^n$, i.e. the linear transformation of $T_p S^n$ that sends a tangent vector $v \longmapsto p \times v$ and $p \times v \longmapsto -v$, being the identity on the orthogonal complement. | |
| Dec 20, 2023 at 7:02 | history | edited | Random | CC BY-SA 4.0 |
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| Dec 19, 2023 at 9:12 | history | edited | Random | CC BY-SA 4.0 |
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| Dec 19, 2023 at 6:25 | history | asked | Random | CC BY-SA 4.0 |