Timeline for Curvature as infinitesimal holonomy 2
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Feb 3, 2021 at 17:39 | comment | added | seub | In other words, the formula is true for general oriented surfaces $S$ with or without boundary, as long as they're compact. | |
| Feb 3, 2021 at 16:11 | history | edited | Sebastian | CC BY-SA 4.0 |
added 189 characters in body
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| Feb 3, 2021 at 15:44 | comment | added | Sebastian | The formula is not true for general surfaces $S$. But if you consider a compact oriented surface $S$ with non-trivial boundary, you can trivialise the principal $S^1$ bundle. This defines a connection 1-form on $S$, also called $\omega$ by abuse of notation, and Stokes theorem gives the holonomy formula. In the case of a closed orientable surface the formula holds as the degree of the $S^1$ bundle is an integer. | |
| Feb 3, 2021 at 14:50 | comment | added | seub | I don't understand: first you provide a "proof" that the formula is true, and then you contradict yourself? The problem with the puncture seems inessential, obviously Stokes doesn't work stricto sensu for noncompact manifolds, so I assume same goes for the formula. Feel free to assume S is compact is my post. | |
| Feb 3, 2021 at 13:59 | history | answered | Sebastian | CC BY-SA 4.0 |