Timeline for Lie groups admitting flat (bi)invariant metrics.
Current License: CC BY-SA 3.0
3 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 30, 2016 at 7:09 | comment | added | Izaak Meckler | What about a screw along an axis? That's an isometry acting without fixed points which isn't a translation. | |
| Jul 13, 2011 at 12:38 | comment | added | Theo Johnson-Freyd | The isomorphism here is as manifolds, not as groups, unless I am very mistaken. There are many nonabelian Lie groups (the solvable connected simply-connected ones) that are isomorphic as manifolds to $\mathbb R^n$. Since the tangent bundle to any Lie group is trivializable by right translations, every Lie group admits a left-invariant metric (pick any metric on the underlying vector space of the Lie algebra). In particular, Alain below gives a noncommutative group with left-invariant flat metric, which is as a manifold just $\mathbb R^3$, but not as a group. | |
| Jul 10, 2011 at 20:40 | history | answered | Igor Belegradek | CC BY-SA 3.0 |