Timeline for $G$-structures of finite type.
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 6, 2011 at 0:39 | vote | accept | Leandro | ||
| May 6, 2011 at 0:00 | comment | added | André Henriques | @Olivier: I think that you're correct. And I think that having a structure of order 3 would mean the following: there should exist a $G$-manifold $M$, and a non-trivial (locally-defined) automorphism around some point $m\in M$ that fixes the 2-jet of that point. | |
| May 5, 2011 at 23:56 | answer | added | Robert Bryant | timeline score: 15 | |
| May 5, 2011 at 23:54 | comment | added | Olivier Bégassat | when you write $\mathbb{R}^n$ do you mean $T_m M$, the tangent space at $m\in M$? Is your notion of $G$ structure the same as that where you ask for an atlas of $M$ such that the transition maps induced on the local trivialisations of $TM$ act like $G$ (for some representation $\roh:G\rightarrow \mathrm{GL}(n)$? | |
| May 5, 2011 at 23:48 | history | edited | André Henriques | CC BY-SA 3.0 |
added 231 characters in body
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| May 5, 2011 at 23:35 | comment | added | Leandro | Actually, while the definition involves the bundle $B_G$, it depends only on the group $G$. You may think of $\mathfrak{g}^{(k)}$ as the space of all $k+1$-multilinear functions $T : (\mathbb{R}^n)^{k+1} \rightarrow \mathbb{R}^n$ which are completely symmetric and such that, given any $v_1, \ldots, v_k \in \mathbb{R}^n$, the linear map $\phi : \mathbb{R}^n \rightarrow \mathbb{R}^n$ given by $\phi(u) = T(v_1, \ldots, v_k, u)$ belongs to $\mathfrak{g}$. | |
| May 5, 2011 at 23:28 | comment | added | Olivier Bégassat | could you explain the meaning of $\mathfrak{g}^{(k)}$? and its relation to $\pi$? | |
| May 5, 2011 at 23:25 | history | asked | Leandro | CC BY-SA 3.0 |