All Questions
Tagged with lie-groups smooth-manifolds
14 questions with no upvoted or accepted answers
7
votes
0
answers
1k
views
Quotient by a non-free action of a Lie group and manifolds with corners
The quotient manifold theorem says that
If $G$ is a Lie group acting freely and properly on a smooth manifold $M$ then $M/G$ has a (unique) smooth structure such that the projection $\pi:M\to M/G$...
- 2,533
6
votes
0
answers
634
views
Quotient space, a fundamental group, and higher homotopy groups 2
Previously, I ask for comments/suggestions on setting up the calculation in Quotient space, homogeneous space, and higher homotopy groups. There, however, I was looking for whatever methods and tools ...
- 10.5k
5
votes
0
answers
135
views
Specify the embedding of Lie groups (via the representation map) precisely as the embedding of two differentiable manifolds
How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$ that agree with a constraint on the mapping between their representations?
By ...
- 10.5k
5
votes
0
answers
157
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Typical preimage of the commutator map
By Goto's theorem for any compact connected semisimple Lie group $G$ of dimension $n$, any element $x\in G$ is a commutator, namely $x=[y,z]$ for some $y, z\in G$. Another way to say it is that the ...
- 728
5
votes
0
answers
477
views
What is the dimension of $M/G$ if it is a manifold and $G$ acts freely and smoothly?
Let $G$ be a Lie group acting smoothly and freely on a smooth manifold $M$. Suppose that the quotient space $M/G$ is a topological manifold. Do we have
$$\dim(M/G)=\dim M-\dim G?$$
Notes: This ...
- 779
4
votes
0
answers
132
views
Coordinates on quotient manifold $\mathrm{SO}(3)/\Gamma$
$\DeclareMathOperator\SO{SO}$Say I have coordinates for $\SO(3,\mathbb{R})$, e.g., a parametrization by Euler angles. Is there a reasonable way to explicitly prescribe coordinates on the quotient ...
- 161
3
votes
0
answers
94
views
The tangent bundle of $G \times_H M$
Let $G$ be a Lie group with a closed subgroup $H$, and let $M$ be a smooth $H$-manifold. I am searching for a reference where it is proved that the tangent bundle of $G \times_H M$ is isomorphic to ...
- 198
3
votes
0
answers
153
views
Diffeomorphisms fixing origin and boundary
Let $D^n$ be a disc in $\mathbb{R}^n$. Is there a known characterization of all the diffeomorphisms of $D^n$ fixing the origin and boundary of $D^n$?
- 5,405
3
votes
0
answers
58
views
Criteria for density of subgroup of diffeomorphism group
Let $C^{\infty,\star}(\mathbb{R}^d)$ denote the non-commutative topological group of smooth diffeomorphisms from $\mathbb{R}^d$ to itself with $\circ$ as multiplication and let $\emptyset\subset X\...
- 5,405
3
votes
0
answers
864
views
Non-Abelian fundamental group? --- a bizarre example
For the quotient space $G=G_0/G_1$, knowing the homotopy
groups of $G_0$ and $G_1$, one can determine homotopy groups from the long
exact sequence
$$
...
\to \pi_n(G_1) \to \pi_n(G_0) \to \...
- 10.5k
3
votes
0
answers
94
views
Cohomology of boundary of locally symmetric space
Let $S$ be a locally symmetric space, not necessarily compact, and $\overline{S}$ be its Borel-Serre compactification. Let $\partial S$ be the boundary of $S$. Let $\widetilde{\mathbb{C}}$ be the ...
- 601
2
votes
0
answers
156
views
Condition on a Lie groupoid to be represented by manifold/group or an action groupoid
Let $\mathcal{G}$ be a Lie groupoid. I am thinking of following questions.
When do we know $\mathcal{G}$ is weakly/Morita equivalent to a Lie groupoid of the form $(G\rightrightarrows *)$ for some ...
- 6,023
1
vote
0
answers
205
views
Fourier transform of functions mapping manifolds, is there a definition?
$\DeclareMathOperator\SO{SO}$I have a problem which boils down to the analysis of functions of the form
$$
f : \mathbb{R} \to \SO(3)^n
$$
Since $\SO(3)$ is a compact group so is $\SO(3)^n$.
Now if ...
- 115
0
votes
0
answers
87
views
Some details about Kirillov-Kostant Poisson bracket
Let $G$ be a finite dimensional Lie group with Lie algebra $\mathfrak{g}$. The Kirillov-Kostant Poisson bracket on $\mathfrak{g}^*$ is defined as $$\{\cdot ,\cdot \} :C^{\infty}(\mathfrak{g}^*)\times ...
- 287