0
votes
2answers
63 views

A full dim. subvector space of $\chi^{\infty}(M)$ which all non zero elements are nonvanishing vec.field

What is an example of a $n$ dimensional manifold $M$ which is not a lie group or $S^{7}$ but satisfies the following property?: There is an $n$ dimensional sub vector space ...
5
votes
2answers
1k views

one-parameter subgroup and geodesics on Lie group

Hi, Given a Matrix Lie Group, I would like to know if the one-parameter subgroups (which can be written as $\exp^{tX}$) are the same as the geodesics (locally distance minimizing curves). Geodesics ...
5
votes
4answers
849 views

The Schwartz Space on a Manifold

I asked this question a couple of days ago on math.stackexchange, but have yet to receive a response, so I have decided to post this here. This question is also vaguely related (both questions arose ...
3
votes
1answer
411 views

Fixed points of the action of an algebraic group

Hello! If a compact Lie group $K$ acts smoothly on a smooth manifold $M$, then the set $M^K$ of fixed points under this action is a smooth submanifold of $M$. This is proved for example in ...
2
votes
3answers
690 views

analytic structure on lie groups

I need a reference for a result I have heard only very vaguely "A lie group (smooth) has a compatible analytic manifold structure". (Would even appreciate a concise way to refer to the result..) I ...
1
vote
1answer
382 views

Understanding manifold GL+(3,R)/SO(3) ?

I'm trying to better understand the manifold GL+(3,R)/S0(3) which is diffeomorphic to positive definite symmetric matrices. My motivation is to understand U in F = RU where F in GL+(3,R) = deformation ...
5
votes
2answers
409 views

Question on transversal slice of Lie group

Assume we have action of Lie group $G$ on a manifold $X$. Fix some orbit $\mathcal{O}$, it is known there exist transversal slice $S$ with respect to this orbit. Fix some point $x$ in $\mathcal{O}$, ...
4
votes
2answers
592 views

Lie Semigroups?

Why is a Lie group wanted instead of a semigroup, what does the group structure give? References on this would be much appreciated. I'm currently pondering manifolds and lie groups and their ...
28
votes
3answers
1k views

What is the classifying space of “G-bundles with connections”

Let $G$ be a (maybe Lie) group, and $M$ a space (perhaps a manifold). Then a principal $G$-bundle over $M$ is a bundle $P \to M$ on which $G$ acts (by fiber-preserving maps), so that each fiber is a ...
10
votes
3answers
838 views

Non-Lie Subgroups

A result of Borel and Lichnerowicz states that the holonomy group of a connection on a principal $G$-bundle is a Lie subgroup of $G$ (Cartan had earlier asserted this, but apparently without proof). ...