Timeline for A variant of the Monge-Cayley-Salmon theorem?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jan 28, 2017 at 22:56 | vote | accept | Terry Tao | ||
| Jan 28, 2017 at 19:01 | comment | added | Robert Bryant | @TerryTao: Yes, they are the left-invariant forms of the affine group. You can think of this group as embedded in $\mathrm{GL}(5,\mathbb{R})$ as the matrix $$g=\begin{pmatrix}1&0&0&0&0\\ x&e_1&e_2&e_3&e_4\end{pmatrix}=\begin{pmatrix}1&0\\x&e\end{pmatrix}.$$ Then the equation $$\mathrm{d}g = g\begin{pmatrix}0&0\\ \omega&\theta\end{pmatrix} = g\,\gamma$$ is the matrix form of the first structure equation, ie., $\gamma = g^{-1}\,\mathrm{d}g$. The second equation is then just $$\mathrm{d}\gamma=-\gamma\wedge\gamma.$$ | |
| Jan 28, 2017 at 18:26 | comment | added | Terry Tao | Thanks for this! I am slowly going through the calculations. Is there a Lie group interpretation of the 1-forms $\omega^i$ and $\theta^j_i$? It looks like they should somehow be associated to the affine group ${\bf R}^4 \rtimes GL(4,{\bf R})$, but I don't see the precise relation yet. | |
| Jan 28, 2017 at 11:50 | history | edited | Robert Bryant | CC BY-SA 3.0 |
Put in a construction of polynomial solutions and corrected some typos
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| Jan 27, 2017 at 21:42 | history | edited | Robert Bryant | CC BY-SA 3.0 |
Fixed some incorrect formulae
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| Jan 27, 2017 at 15:17 | comment | added | Deane Yang | Very nice application of exterior differential systems. | |
| Jan 27, 2017 at 14:31 | history | edited | Robert Bryant | CC BY-SA 3.0 |
added 8451 characters in body
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| Jan 27, 2017 at 14:23 | comment | added | Robert Bryant | @TerryTao: Ok, I've put it in. If you have questions, or something needs clarification, please let me know. | |
| Jan 27, 2017 at 3:22 | comment | added | Terry Tao | I would be interested in seeing more details, thanks. Right now I don't see how the bundle $F$ is interacting with the subspaces $W$. | |
| Jan 27, 2017 at 2:18 | history | answered | Robert Bryant | CC BY-SA 3.0 |