Timeline for Splitting Short exact sequences of vector bundle with connection
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Oct 1, 2016 at 10:49 | comment | added | Holonomia | Maybe I got lost with the terminology, but it seems to me that if 1) holds then in each fiber of $F$ there is a complement of $E$ invariant by the holonomy group. So now it is enough to know examples of linear connections whose holonomy group have invariant subspaces without invariant complements e.g. Lorentzian geometry, take the holonomy group if a indecomposable but not irreducible Lorentzian manifold. If the connection is flat then the problem becomes a topological one i.e. if $M$ is connected and simply connected then 1) is true under the flatness of $\nabla$. | |
| Oct 1, 2016 at 0:50 | comment | added | Alex Degtyarev | 3: Also, it seems to me that, in the Riemennian case, the splitting is given by the orthogonal complement. | |
| Oct 1, 2016 at 0:06 | comment | added | Alex Degtyarev | 2: flat connection $=$ representation of $\pi_1$. If the group is infinite, there's no reason why the category should be semisimple. (Just take a nontrivial Jordan block.) | |
| Sep 30, 2016 at 23:47 | review | First posts | |||
| Oct 1, 2016 at 0:07 | |||||
| Sep 30, 2016 at 23:46 | history | asked | Miquel | CC BY-SA 3.0 |