Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question

2 Answers 2

up vote 2 down vote accepted

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).

share|improve this answer

See page 65 of here.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.