Timeline for pull-back connection
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 12, 2019 at 12:50 | comment | added | Alex M. | @user11538: I don't think this answer is correct: in order for that tensor product to make sense, how can $\Gamma(P, \mathbb C)$ be a module over $\Gamma(B, \mathbb C)$? In other words, how can you multiply a function on $P$ by a function on $B$ and obtain a function on $P$? | |
| Dec 14, 2010 at 0:01 | comment | added | BCnrd | Dear unknown: later in life you'll want to use complex manifolds or complex algebraic varieties. (Connections remain all about making vector fields act as "directional deriv." operators on sections of the bundle, as in diff'ble case.) Then no bump functions, so cannot work entirely so "globally" as above. That's why I outlined an alternative to exploit your preferred calculations on a local level, coupled with global uniqueness (which can be proved by local calculations!) to infer global existence. It really is easier that way (e.g., no non-obvious isoms needed). You'll appreciate it later. | |
| Dec 13, 2010 at 20:30 | comment | added | user11538 | Dear BCnrd, Thank you for your interesting comments but... 1) I was only interested in the differentiable case. I am not sure what you mean by a connection in the analytic or algebraic cases. (an analytic(algebraic) splitting of the tangent bundle?) so I am really satisfied with the isomorphism of smooth sections i was talking about. By the way, it's not mine. 2)I admit the comment to Matt lacked details, probably thinking that it was clear $X$ should be thought as a tangent vector and not a vector field (as I wrote); (retrospectively that was really not the main issue of the question); | |
| Dec 13, 2010 at 18:26 | comment | added | BCnrd | Dear unknown: the isom. you're using is false in complex-analytic and algebraic cases (unsure in $C^\infty$-case, but seems silly to use a defn that only works there when there's a simple procedure applicable in all cases). Think more locally (it's good for you!): use local existence and global uniqueness to get global existence. What you propose with "global" tensors works locally with no hard work; then use uniqueness to globalize. Your comment to Matt is unclear since your original statement of the pullback-relation with global vector fields makes no sense ($dF(X)$ makes no sense on $B$). | |
| Dec 13, 2010 at 17:29 | comment | added | user11538 | Willie i didn't say it was easy:) In the background lurks the isomorphism $\Gamma(P;\pi^*E)\simeq \Gamma(P;\mathbb{C})\otimes \Gamma(B;E)$ which is by not obvious(at least not to me). But Leibniz relation does make sense and it does define the connection everywhere and it also explains the trivial connection on the trivial bundle. | |
| Dec 13, 2010 at 17:15 | comment | added | Willie Wong | @Unknown: see my comment above. The equation you wrote down is only meaningful for pull-backs of sections $s$. If you multiply $F^*s$ by an arbitrary function $h$, it will no longer be the pull back of a section over $B$ if $h$ is not constant along $F^{-1}B$. In other words, you were trying to force Leibniz rule somewhere it has no business being. The given expression is, indeed, enough to specify the pull-back connection, but it is not so simple as plug-and-play. | |
| Dec 13, 2010 at 17:10 | history | answered | user11538 | CC BY-SA 2.5 |