1
$\begingroup$

Let $G$ be a Lie group and $H$ a closed Lie subgroup. Is there an explicit way to construct a local cross section of $H$ in $G$ so that $\pi: G\to G/H$ is a fiber bundle?

Improve this question
$\endgroup$
1
  • $\begingroup$ Do you have any follow-up work on constructing the local cross section? $\endgroup$
    – Troy Woo
    Commented Feb 6, 2015 at 11:25

2 Answers 2

Reset to default
3
$\begingroup$

You might consider your problem in a generalized setting. Let $H$ be a Lie group and $X$ an $H$-manifold on which $H$ acts freely and properly. There is a unique manifold structure on $X/H$ for which $X\rightarrow X/H$ is a principal $H$-bundle (and in particular, a fiber bundle). If $H$ is a closed subgroup of a Lie group $G$, then $H$ acts freely and properly on $G$ by left-multiplication. You then obtain the desired result.

I have never before attempted to construct explicit local cross-sections. You might try to use the exponential map $\exp:\frak{g}\rightarrow $G$ $ (a local diffeomorphism) together with your surjection $\frak{g}\rightarrow\frak{g}/\frak{h}$ to construct a local section at the identity. Otherwise, I think you might need to consider the Implicit Function Theorem (not particularly explicit).

Improve this answer
$\endgroup$
3
$\begingroup$

See page 65 of here.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Nice book. You should have sent me the link when you answered my question on math.SE :-) $\endgroup$
    – Troy Woo
    Commented Feb 6, 2015 at 11:26

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .