Timeline for Maximal symmetry and isometries not connected to the identity
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Jan 21, 2017 at 8:11 | comment | added | Ben McKay | If your Killing vector fields are complete, then they generate a group action by a transitive group of isometries. Constant curvature (as above) implies isomorphism with a space form. So your space form examples are the only examples. | |
| Jan 21, 2017 at 0:26 | comment | added | Guillem Pérez-Nadal | Thanks Ben for your answer. What I mean by a Killing field is a vector field which satisfies the Killing equation and which is complete, so that it generates a one-parameter group of isometries. Perhaps I should have emphasized that. This excludes open subsets of Euclidean space as examples of maximally symmetric spaces. | |
| Jan 20, 2017 at 22:09 | comment | added | Ben McKay | To give more detail, break up the curvature tensor into its irreps. Then all of these vanish, except for the trivial representation (corresponding to the scalar curvature), so the traceless Ricci and the Weyl vanish. | |
| Jan 20, 2017 at 22:05 | comment | added | Ben McKay | To be very specific, take as your open subset of Euclidean space the interior of a nonisoceles triangle. | |
| Jan 20, 2017 at 22:03 | history | answered | Ben McKay | CC BY-SA 3.0 |