I am curious if the set of all orientation-preserving diffeomorphisms with a given rotation number is a tame Lie subgroup or a tame submanifold of all orientation-preserving diffeomorphisms on the unit circle? Thanks.
4
-
$\begingroup$ The subset of diffeomorphisms with a given rotation number $n$ is never a subgroup except if $n=0$ since you add rotation numbers when you compose diffeomorphisms. $\endgroup$– Tobias DiezCommented Mar 7, 2018 at 9:57
-
$\begingroup$ Yes, I agree, but is it a tame submanifold? Or, if you consider torus diffeomorphisms isotopic to the identity, is that a tame Lie subgroup of the group of difeomorphisms? $\endgroup$– Hua YingCommented Mar 7, 2018 at 10:01
-
$\begingroup$ It should be a smooth submanifold, since the rotation number gives you a smooth group homomorphism onto $\mathbb{Z}$, which is hence automatically a submersion. Concerning the torus case, do you mean the connected component of the identity? These are automatically Lie subgroups (this fact is probably shown in arxiv.org/abs/1501.06269). $\endgroup$– Tobias DiezCommented Mar 7, 2018 at 10:45
-
$\begingroup$ Thank you very much for your comments. Yes, I mean the connected component, so in that case we can identify its tangent spaces with those of the whole group of diffeos? Isn't the rotation number homomorphism onto the unit circle group? $\endgroup$– Hua YingCommented Mar 7, 2018 at 11:00
Add a comment
|