Timeline for Flat connections and global sections of vector bundles
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 14, 2019 at 0:53 | comment | added | Deane Yang | Simply define a global section by setting the value at any point $y$ as the parallel transport of $v_0$ along any smooth curve from $x$ to $y$ and observe that if the connection is flat then any two smoothly homotopic curves give the same value at $y$. | |
| Oct 13, 2019 at 18:47 | comment | added | Ron | @SamGunningham Thank you. This helps. Can you suggest a good reference for this. | |
| Oct 13, 2019 at 18:28 | comment | added | Sam Gunningham | In general, flat connections on a manifold (in either the C^\infty or holomorphic setting) are in correspondence with representations of the fundamental group. The correspondence takes the bundle $V$ to its fiber $V_x$ at the basepoint, equipped with the action of $\pi_1$ via parallel transport. In these terms, you can see that one can lift $v\in V_x$ to a global flat section if and only if $v$ is invariant under the action of $\pi_1$. | |
| Oct 13, 2019 at 18:19 | comment | added | Praphulla Koushik | I am reading your comment as "because of flat connection, one can define parallel transport".. Is this what you mean? For any connection on vector bundle we can talk about parallel transport.. did I misunderstand some things | |
| Oct 13, 2019 at 17:41 | history | edited | Ron | CC BY-SA 4.0 |
added 48 characters in body
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| Oct 13, 2019 at 17:40 | review | Close votes | |||
| Oct 26, 2019 at 23:27 | |||||
| Oct 13, 2019 at 17:35 | comment | added | Ron | @PraphullaKoushik non-singular is the same as smooth in this setup. The main point is the existence of a flat connection (not all vector bundles can be equipped with a flat connection), which enables us to define parallel transport. However, I am not sure if one can use this parallel transport to define a global section. | |
| Oct 13, 2019 at 17:29 | comment | added | Praphulla Koushik | Let $M$ be a smooth manifold and $E\rightarrow M$ be a vector bundle. Let $\nabla$ be a connection on the vector bundle $(E,\pi,M)$. Fix a point $m\in M$ and an element $v$ in the fibre of $m$... Does there exists a global section $s:M\rightarrow E$ such that $s(m)=v$?? | |
| Oct 13, 2019 at 17:25 | comment | added | Praphulla Koushik | Do you think same question in setting of smooth manifolds have same answer? | |
| Oct 13, 2019 at 17:07 | history | asked | Ron | CC BY-SA 4.0 |