Timeline for Flat R-bundles on surfaces
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 15, 2018 at 18:49 | comment | added | Ben McKay | Calculate for yourself as a simple example the parallel transport for the trivial rank 1 bundle on the circle, with nonzero constant connection coefficient. You can see that the parallel transport is not the identity (indeed it is exponential growth), so there is no global parallel nonzero section. | |
| Aug 15, 2018 at 16:20 | vote | accept | BiM | ||
| Aug 15, 2018 at 16:12 | comment | added | BiM | Hah, Thanks, anther point of my question need to be clarify, anyway, so the answer is those vector bundle always can be trivializable. I do not understand your words ´´not by parallel sections´´, then generally, how to trivialize such kind of vector bundle without using parallel sections? | |
| Aug 15, 2018 at 15:55 | comment | added | Ben McKay | Your question was whether the flat $\mathbb{R}^+$-bundles are trivial. They are not. But as you say the associated vector bundle is smoothly trivializable, but not by parallel sections, for any representation of $\mathbb{R}^+$. | |
| Aug 15, 2018 at 15:47 | comment | added | BiM | This homology group actually corresponding to the change of flat connections, but the underlying vector bundle is the same trivial bundle. | |
| Aug 15, 2018 at 15:37 | history | answered | Ben McKay | CC BY-SA 4.0 |