Timeline for About the Cartan's moving frame method
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 16, 2022 at 15:23 | comment | added | A. J. Pan-Collantes | @DeaneYang I realize this is difficult to deal with in comments, so I have created a new question: mathoverflow.net/questions/432563/… | |
| Oct 15, 2022 at 17:57 | comment | added | A. J. Pan-Collantes | @DeaneYang Ok with respect to the no inclusion of points. WRT the generality of the construction: if the space is a homogeneous space $X=G/H$ then $G$ is a frame bundle for $X$ and $G$ itself acts faithfully and transitively on it (isn't it?). But this construction of Griffiths or Clelland doen't have to work here, we only can write the MC form "in abstract". I guess you have to require something else (for example $G=\mathbb{R}^n \rtimes H$ and $\mathfrak{g}=\mathbb{R}^n \oplus \mathfrak{h}$?) | |
| Oct 10, 2022 at 19:30 | comment | added | Deane Yang | @AntonioJPan, it is important to note that points are not contained in $E(N)$. Instead, there is a map from the frame bundle to the space of points. This construction works if there is a frame bundle over a space and a Lie group that acts faithfully and transitively on the frame bundle. | |
| Oct 10, 2022 at 6:16 | comment | added | A. J. Pan-Collantes | @J.Doe I explain a bit more: I know that MC-form is a general construction for any Lie group. But the general way we construct them in any Lie group (fixing a basis of left invariant vector fields) is not what is done in Griffiths or Clelland. Is this particular construction only valid in a few situations? | |
| Oct 10, 2022 at 6:02 | comment | added | A. J. Pan-Collantes | @DeaneYang With respect to the first comment I think (but I want to be confirmed) that the point is not that $\mathbb{R}^N$ "is contained" in $E(N)$, but that the elements of $E(N)$ can be "characterized" by $(N+1)$-uples of elements of $\mathbb{R}^N$: $(p,e_1,...)$. Together with the fact that $T_pR^N=R^N$. But then I have a question: this is not a valid construction for every Lie group, or it is? | |
| Jul 16, 2019 at 2:33 | comment | added | Deane Yang | @JDoe, I think Ben McKay's recommendation of Jeanne Clelland's book is a good one. | |
| Jul 15, 2019 at 22:22 | vote | accept | J.Doe | ||
| Jul 15, 2019 at 22:18 | comment | added | J.Doe | Can you expain how to show that the forms $\psi_{\alpha}$ in $d\psi(F)=\displaystyle{\sum_{\alpha=1}^{N}}\psi_{\alpha}(F)e_{\alpha}$ are left invariant on $E(N)$ on p.783 of Griffiths's paper ? and can you recommand some reference for Lie groups and homogeneous space and the principal bundles (concerning this problem)? | |
| Jul 15, 2019 at 22:00 | comment | added | J.Doe | I have read Griffiths' paper. I think the point is that we treat $\mathbb{R}^N$ as the quotient space of the Lie group $E(N)$ (which is isomorphic to the frame bundle) under the action of $O(N)$ so that the ambient space (i.e. $\mathbb{R}^N$) of the functions $\psi:\,E(N)\rightarrow\mathbb{R}^N$ (p 783 of Griffiths paper) can be contanined in $E(N)$ so that the equation $d\psi(F)=\displaystyle{\sum_{\alpha=1}^{N}}\psi_{\alpha}(F)e_{\alpha}$ makes sense. Am I right? | |
| Jul 15, 2019 at 16:03 | history | answered | Deane Yang | CC BY-SA 4.0 |