Timeline for Connection Transformation Formula; Degree 3 Cech Cohomology
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 7, 2012 at 8:11 | comment | added | Peter Michor | Let $\mu:G\x G\to G$ be the multiplication and let $x.y = \mu(x,y) = \mu^y(x) = \mu_x(y)$; this is to fix notation for right and left transport on $G$. Then, for $f:M\to G$, the notation $f^{-1}.df$ is shorthand for `$$x\mapsto T(\mu_{f(x)^{-1}).T_x f: T_xM \to T_{f(x)} G \to T_eG = \mathfrak g$$' This also works in infinite dimensions. | |
| Dec 7, 2012 at 1:03 | comment | added | cheyne |
Also, I don't understand your "definition" of $f^*\omega$ = f^{-1} \omega (df)$, when $f$ is $G$-valued and $\omega$ is $\mathfrak{g}$ - valued; unless you implicitly mean an adjoint map induced by $f$ on the Lie-Algebras?
|
|
| Dec 7, 2012 at 0:46 | comment | added | cheyne | Thank you very much for this, Johannes. I am working through the details at the moment. Firstly, I think the main thing I was missing was that any connection can be decomposed where one piece is the Maurer Cartan form; this is obvious to me now (thanks!). Secondly, I am working through your equations after "Next you have to invoke the fact" and I am just making sure that when you talk about "linear groups" you are allowing for infinite dimensions? | |
| Dec 7, 2012 at 0:07 | history | answered | Johannes Ebert | CC BY-SA 3.0 |