Timeline for Curvature of principal bundle
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 27, 2022 at 9:17 | comment | added | Alex | In the last line, you need to show that $[X^V, Y^V]$ and $[X^V, Y^H]$ are vertical, right? | |
| May 13, 2021 at 14:22 | comment | added | NicAG | Sorry, I have to digest all these definitions. But it essentially means, that the curvature of $P$ is defined by the connection and the curvature of $M$. Since the horizontal subspace $T_pH$ is isomorphic to $T_pM$, the definition above tells us how much $T_pM$ and $T_qM$ differ. Or how much the actual connection of $T_pM$ and $T_qM$ differs from the flat connection | |
| May 13, 2021 at 13:17 | comment | added | NicAG | Thanks. Then there is maybe a mistake in the wikipedia article, because they also use the $1/2$ in the structure equation. | |
| May 13, 2021 at 11:41 | comment | added | HYL | There are two conventions defining $d \omega$ for a $k$-form $\omega$, according to whether there is a factor $\frac{1}{k+1}$ or not. In Kobayashi-Nomitsu's book, the factor $\frac{1}{k+1}$ is in the definition, this is why we have $\frac{1}{2}$ in the structure equation. | |
| May 13, 2021 at 11:34 | comment | added | NicAG | @HYL Thanks. That's what I was looking for. I am just a bit confused. The expression for $\Omega$ then needs a $-1/2$ factor which is missing in wikipedia. | |
| May 13, 2021 at 11:31 | vote | accept | NicAG | ||
| May 13, 2021 at 11:26 | comment | added | NicAG | @მამუკაჯიბლაძე It is a 2 form. I just wrote down the general expression for $D_{\omega} \omega$ if $\omega$ is a $k$-form | |
| May 13, 2021 at 6:46 | comment | added | მამუკა ჯიბლაძე | But $D_\omega\omega$ of the OP seems to be not a 2-form? As written, it has values on $k$-tuples of vector fields for all $k$ | |
| May 13, 2021 at 6:27 | history | answered | HYL | CC BY-SA 4.0 |