Timeline for Normalizing the value of a principal connection at a point
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 27, 2014 at 16:26 | vote | accept | José Navarro | ||
| Jan 27, 2014 at 16:25 | comment | added | José Navarro | You are all right: I'm not used to this language yet and, by trying to simplify the question, I got confused. What I was trying to argue was the existence of a section $\sigma \colon X \to P_0$ such that $\alpha_{|\sigma}$ vanishes at $p$, and any of your reasonings is enough. | |
| Jan 27, 2014 at 13:02 | comment | added | Sebastian | You both are right, I was not clear enough with my phrasing "connection 1-form with respect to this section vanishes at p," which should mean pullback of the connection form by the section. And Robert's remark shows us how nicely differential calculus can be approximated linearly, I just oversaw it. | |
| Jan 27, 2014 at 12:24 | comment | added | Robert Bryant | To find such a section, you don't need to do anything with parallel transport (which involves solving differential equations). It suffices to take any section $\sigma:X\to P_0$ such that the tangent space of the image $\sigma(X)\subset P_0$ at $(g,x)$ is equal to the kernel of $\alpha_{(g,x)}:T_{(g,x)}\to{\frak{g}}$, for then $\sigma^*(\alpha)$ will vanish at $x$, which is what you really want (instead of $\alpha_{(g,x)}$ vanishing, which, as Oldfich noted in his comment above, never happens). | |
| Jan 27, 2014 at 11:57 | comment | added | Oldřich Spáčil | I guess by $p$ you mean a point on the base $X$ and you want to say that the pullback of the connection form $\alpha$ along your local section vanishes at $p$? | |
| Jan 27, 2014 at 11:09 | history | answered | Sebastian | CC BY-SA 3.0 |