2
votes
1answer
33 views

quotient by a group that acts almost freely

How can I show that if a compact lie group G acts almost freely and smoothly on a manifold M, then M/G is Hausdorff? (an action is almost free if $G_x$ is finite for all x $\in$ M)
1
vote
2answers
73 views

Action of $SO_n$ on $\mathbb{S}^{n-1}$ induces fibre bundle.

Real compact Lie group $SO_n$ acts smoothly and transitively on $\mathbb{S}^{n-1} \subseteq \mathbb{R}^n$ with obvious action. Isotropy subgroup of each point in $\mathbb{S}^{n-1}$ is isomoprhic to ...
6
votes
2answers
91 views

Eigenbundle decomposition

Let $G$ be a finite cyclic group and $X$ a smooth manifold equipped with a trivial $G$-action. It is known that we can decompose every $G$-equivariant vector bundle with respect to the action: ...
1
vote
0answers
44 views

invariance of 2-form under $SO(3)$

I'm trying to understand how to derive forms that invariant under action of some group. For example 2-form on $S^2$ and on $\mathbb{R}^3$ is very interesting for me (because I have troubles with it). ...
11
votes
1answer
271 views

The Quaternions and $SO(4)$

I am interested in the map $\phi:S^3 \times S^3 \to GL_4(\mathbb{R})$ given as follows: Let $(p,q) \in S^3 \times S^3$. We identify $p$ and $q$ as real quaternions with unit norms and define ...
1
vote
0answers
45 views

Exponential intertwining of linear and local actions.

I am reading Duistermaat's 1973 paper on relating the convexity of the image of a moment map to the image of the fixed points of an antisymplectic involution. In that paper, the following comment is ...
2
votes
0answers
71 views

An orbit of a group action and the implicit function theorem

Suppose that a Lie group $G$ acts smoothly on a manifold $X$. We can easily prove the following theorem by using the constant rank theorem, which is a stronger theorem than the implicit function ...
4
votes
0answers
171 views

Group actions and associated bundles

Let $P$ be a principal $G$-bundle over $B$, and let $G$ act on some space $F$ (feel free to work in your favorite category of spaces, if this helps). Then $\text{Aut}{P}$ (aka the group of gauge ...
7
votes
1answer
215 views

Is the Structure Group of a Fibre Bundle Well-Defined?

Am I right in thinking that the structure group of a fibre bundle is any group $G$ of homeomorphisms of the fibre $F$ such that all transition functions map into $G$? Or is $G$ somehow the minimal ...
1
vote
2answers
67 views

Finiteness of fixed points of a Lie group action

Let $\psi: G\rightarrow \mathrm{Diff}(M)$ be a smooth non-trivial action of a compact connected Lie group $G$ on a compact connected smooth manifold $M$. Under which assumptions there will be a ...
1
vote
0answers
49 views

A K-invariant submanifold of G-manifold and fundamental vector fields

Let a (connected) Lie group $G$ act on $M$. Assume that the action is locally free. (In other words, if the fundamental vector field of $X \in \mathrm{Lie(G)}$ $$ \underline{X}(p) := ...
1
vote
1answer
69 views

Definition of the rank of the isometry group of a manifold.

Let $M$ be a manifold (which you can assume compact if it helps) and consider the natural action of the isometry group of $M$, Iso($M$) on $M$. The we can define the symmetry rank of $M$ as the rank ...