Timeline for Existence of connections making a bundle endomorphism parallel
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 1, 2013 at 9:50 | comment | added | Ben McKay | The Nijenhuis tensor of $N$ also has to vanish. (So the notation $N$ is perhaps not ideal here.) | |
| Sep 1, 2013 at 9:39 | comment | added | Ben McKay | But then the generalized eigenspaces are also parallel: $(N-\lambda I)^k=0$. By torsion-freedom, that should make them Frobenius. | |
| Sep 1, 2013 at 9:31 | comment | added | Mariano Suárez-Álvarez | Yup, decomposable endomorphisms do not work, but I am having trouble breaking this with one Jordan block. | |
| Sep 1, 2013 at 9:28 | comment | added | Ben McKay | Robert wakes up early. | |
| Sep 1, 2013 at 9:27 | comment | added | Robert Bryant | That's not enough either. For example, if $N$ has exactly two eigenvalues $1$ and $-1$ and they are of constant multiplicity, then there will exist a torsion-free connection making $N$ parallel if and only if the two eigenbundles in $TM$ are Frobenius. | |
| Sep 1, 2013 at 9:26 | comment | added | Mariano Suárez-Álvarez | Suppose then the Jordan type is constant :-) | |
| Sep 1, 2013 at 9:18 | history | answered | Ben McKay | CC BY-SA 3.0 |