Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Question 1:

Let $G$ be a Lie group and $H<G$ a Lie subgroup of $G$, given the projection $p:G\to G/H$ and a smooth (local) section $s:G/H\to G$ s.t. $p\circ s=id$, then does this imply that $s$ is an immersion? So a section can in general be considered a submanifold of $G$? When will $s$ be an embedding?

Question 2:

If the first case is true, denote the image of $s$ by $M$. Then is it true that the pointwise multiplication $M\cdot H=G$ (at least locally)? On the other hand, if $M$ is given that satisfies $M\cdot H=G$, is it true that $M$ is always the image of a smooth section? Is there a standard way to construct and classify submanifolds of $G$ that satisfies $M\cdot H=G$?

share|improve this question

migrated from mathoverflow.net Jul 13 '13 at 13:33

This question came from our site for professional mathematicians.

1 Answer 1

up vote 3 down vote accepted

Question 1: $p:G\to G/H$ is a principal fiber bundle with structure group $H$. Thus $s$ is an immersion (since $p\circ s =Id$, $s$ has rank $=\dim(G/H)$, and $s$ is injective and a closed embedding).

Question 2: Yes, $M.H=G$. But not every $M$ with this property is the image of a section. $M$ might be a leaf of the horizontal foliation given by a flat principal connection, then $p:M\to G/H$ would be a covering mapping.

share|improve this answer
    
Please also consider my last subquestion... –  Troy Woo Jul 13 '13 at 14:08

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.