All Questions
Tagged with lie-groups group-actions
65 questions
10
votes
1
answer
605
views
Is it possible to average a riemannian metric over an action and preserve curvature bounds?
Let $M$ be a finite dimensional smooth manifold endowed with a riemannian metric $g$ and a smooth action $\mu$ by a compact Lie group $G$. Averaging $g$ over $G$ defines a new metric
$$g'(X,Y)=\int_Gg(...
- 45k
3
votes
0
answers
134
views
When do quotients of $G$-vector bundles exist?
Let's work in the category of smooth (paracompact, Hausdorff) manifolds. Let $M$ be a manifold and $G$ a Lie group acting on $M$. Suppose $E$ is a $G$-vector bundle on $M$ (that is, $G$ acts on $E$ by ...
- 51
0
votes
0
answers
126
views
Expressing the union of principal orbits as a disjoint union of global slices for proper group actions
Setup:
I was reading about slices and principal orbit theorems (Theorem 3.4.6) from these notes.
Let the Lie group $G$ act on a complete Riemannian manifold $(M,g)$ isometrically on $M$, i.e. $\phi^{*}...
- 1,467
1
vote
0
answers
67
views
Is the adjoint action of $\mathrm{SU}(2)$ on the Schwartz space proper and free?
$\DeclareMathOperator\SU{SU}$Let $t_1, t_2, t_3$ be generators of the Lie algebra $\mathfrak{su}(2)$.
Let us consider a Schwartz space $\mathcal{S}$ defined as
\begin{equation}
\mathcal{S}:= \Bigl\{ \...
- 63.9k
2
votes
0
answers
45
views
Topology of an orbit space constructed from a Fréchet space under the "local" action of some "smooth" group
Let $G$ be a nontrivial connected compact subgroup of the general linear group $\operatorname{GL}(\mathbb{R}^3)$. For example, we may take $G$ to be $\operatorname{SO}(3)$.
Next, let $\mathcal{S}(\...
- 12.9k
4
votes
1
answer
235
views
If a discrete and faithful representation of a surface group has proximal values, does the attracting points map have a continuous extension?
For some context, I'm studying the paper Anosov Representations and Proper Actions [GGKW].
$G$ denotes a non-compact real reductive Lie group of rank greater than $1$, $\Gamma$ denotes the fundamental ...
7
votes
1
answer
178
views
Homogeneous metric connections on 3-dimensional Lie groups
Let $G$ be a 3-dimensional unimodular Lie group equipped with a left-invariant metric $q$. Call $P_{SO}$ its oriented orthonormal frame bundle.
Considering the moduli space of connections $\mathscr{B}$...
- 108k
1
vote
0
answers
115
views
A compact Lie group $G$ acting on a compact Lie group $K$ transitively. Is there a $C$ such that $d(gx,gy)\leq Cd(x,y)$?
Let $G$ be a compact connected Lie group acting transitively and smoothly on another compact Lie group $K$. Let $d$ be the distance in $K$ that is not $G$-invariant. Is there a constant $C$ such that $...
- 169
4
votes
1
answer
149
views
An analogue of Mostow-Palais equivariant embedding theorem for the group of conformal automorphisms of the 2-sphere
Is there a smooth embedding of $S^2$ into some Euclidean space that is equivariant with respect to a linear representation of $PSL(2,\mathbb C)$?
A counterexample to a more general question can be ...
- 108k
0
votes
0
answers
97
views
Methods for calculating (one-parameter subgroup) actions
For $G$ a Lie group and $\mathfrak{g}$ its Lie algebra, I am interested in one-parameter subgroup actions on “functions” $f$ of the form
\begin{equation}
\mathrm{e}^{t L(z)} f(z)
\end{equation}
...
- 63.9k
2
votes
1
answer
136
views
Generalization of $G/T \simeq G_\mathbb{C}/B$
Let $G$ be a compact Lie group and Let $G_\mathbb{C}$ be its complexification. Let $T$ be a maximal torus of $G$ and let $X$ be the quotient $G/T$.
Consider $H$ to be a Lie subgroup of $G$ and denote ...
- 148k
1
vote
0
answers
48
views
Relation between weight spaces of fixed loci of Hamiltonian $S^1$-actions
Consider an almost Kähler manifold $(M,\omega,I)$ with a $I$-(pseudo)holomorphic $\mathbb{C}^*$-action, whose $S^1$-part is Hamiltonian and the fixed locus
$F=M^{S^1}$ is compact. Then, it breaks $F=\...
- 1,677
5
votes
1
answer
252
views
Does a compact Lie group action on a family of compact manifolds have diffeomorphic fixed point submanifolds?
Let $\pi: M\to B$ be a fiber bundle of smooth manifolds with $B$ connected and each fiber of $\pi$ is a compact manifold. Let $G$ be a compact Lie group acting smoothly on $M$ such that
$\pi(g\cdot m)=...
- 1,908
2
votes
0
answers
89
views
Question about finite dimensional representations of a semi-simple Lie group
I have posted a question in MSE
https://math.stackexchange.com/questions/4468138/question-about-finite-dimensional-representations-of-a-semi-simple-lie-group but didn't receive any comment or answer.
...
- 139
7
votes
1
answer
394
views
On fixed point sets of actions of compact Lie groups
Let a compact Lie group $G$ act smoothly on a compact smooth manifold $M$. For any compact subgroup $H\subset G$ denote by $E^H$ the image in $M/G$ of the fixed point set of $H$ in $M$.
Is it true ...
- 63.9k
3
votes
1
answer
337
views
Path lifting property for $\pi:M\rightarrow M/G$ for $G$ compact Lie acting smoothly and freely
Let $M$ be a smooth manifold and let $G$ be a compact Lie group acting smoothly and freely over $M$. Let $\pi:M\rightarrow M/G$ be the canonical projection, and endow $M/G$ with the unique ...
- 8,529
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 ...
- 63.9k
3
votes
1
answer
105
views
Define a symplectic structure on $G \times_{G_\beta} V$, where $V$ is symplectic
Let $G$ be a compact Lie group with algebra $\mathfrak{g}$. Let $\beta $ be an element in the dual of the Lie algebra $\mathfrak{g}$. We denote by $G_\beta$ the stabilizer subgroup of $\beta$ by ...
- 8,597
3
votes
1
answer
392
views
Almost free Lie group action
It's known that if a compact Lie group $G$ acts freely on a compact manifold $M$, then the orbit space $M/G$ is a manifold. If we only assume that $G$ acts almost freely (i.e. $G_x$ is finite for any $...
- 1,908
8
votes
0
answers
285
views
Fundamental domains for proper Lie group actions on smooth manifolds
The setting: $M$ an arbitrary smooth manifold, $G$ a Lie group acting effectively and properly on $M$ by diffeomorphisms.
Motivation: when trying to figure out the homeomorphism type of the orbit ...
- 183
3
votes
0
answers
86
views
non-smooth manifold with circle action (with fixed points)
I am interested to know if there a non-smooth manifold (i.e. a closed topological manifold admitting no smooth structure) $M$, having a continuous action $M \times S^1 \rightarrow M$, and the number ...
- 6,995
21
votes
5
answers
3k
views
How to compute the (co)homology of orbit spaces (when the action is not free)?
Suppose a compact Lie group G acts on a compact manifold Q in a not necessarily free manner. Is there any general method to gain information about the quotient Q/G (a stratified space)? For example, I ...
- 12.9k
3
votes
0
answers
74
views
Coordinates for quasiperiodic motion after reconstruction
Consider a free action of $SO(3)$ on a manifold $M$ and some (reducible) dynamics vector field $X$ on $M$. Suposse that the reduced dynamics $X_{red}$ on $M/SO(3)$ has only fixed points and periodic ...
- 141
4
votes
2
answers
419
views
Three dimensional real Lie groups with cocompact discrete subgroups
I would like to know what are all the real three dimensional Lie groups (simply connected) that can act transitively and locally freely on a compact three dimensional manifold?
This is equivalent to ...
- 63.9k
1
vote
0
answers
91
views
Is this equivariant function constant?
Let $G$ be a linear algebraic group (think of $SL_n(\mathbb{R})$), $B$ its Borel (standard minimal parabolic) subgroup (think of upper triangular subgroup), and let $\Gamma \leq G$ be a cocompact ...
- 949
8
votes
1
answer
298
views
How special are homogeneous spaces?
Let $M$ be a smooth finite dimensional manifold, how restrictive is it to require $M$ to admit a smooth action by a finite dimensional Lie group $G$?
Related questions/approaches: Of course we need $\...
- 8,098
2
votes
1
answer
193
views
Why does the formula (7) in the article equivariant cohomology with generalized coefficients hold
I'm reading the article of equivariant cohomology with generalized coefficients by Kumar and Vergne and I have this question from that article
Let G be a compact lie group with lie algebra $\mathfrak{...
- 148k
5
votes
1
answer
222
views
Fixed sets of orbit spaces
I've run across something that surprises me, so I'm wondering (1) Is it true? and (2) Is it well known? (And if the answers are affirmative, why didn't I know this already?)
Let $G$ be a compact Lie ...
- 2,062
3
votes
0
answers
83
views
Quotients of a fixed manifold by a fixed Lie group
Let $M$ a connected paracompact differentiable manifold. Let $G$ a connected Lie group. I am interested in the possible "regular" (e.g. smooth) quotients of $M$ by actions of $G$. What ...
- 141
1
vote
0
answers
51
views
Identifications of the space $A^* (G \times _H M) $
Let $G$ be a lie group, and let H be lie subgroup of G acting on G by right translation.
Let M be a H-manifold.
Why do these equalities hold:
For the algebra $ A^* (G \times _H M)$, we have:
$ A^* (G \...
- 133
7
votes
1
answer
279
views
Question about an example in symplectic geometry
Let M be a coadjoint orbit of dimension 6 of $SU(3)$, and let T be the maximal torus in $SU(3)$. If we denote $\mu : M \longrightarrow \mathbb{R}^2$ the moment map associated to the action of T on M, ...
- 12.9k
8
votes
1
answer
305
views
Groups that act transitively on $\mathrm{Gr}(k,\Bbb R^n)$ but not transitively on $\mathrm{Gr}(k+1,\Bbb R^n)$
Is it known for which $n, k\in\Bbb N$ there exists a matrix group $\Gamma\subseteq\mathrm{GL}(\Bbb R^n)$ that
acts transitively on $\mathrm{Gr}(k,n)$, i.e., on the $k$-dimensional subspaces of $\Bbb ...
- 108k
19
votes
1
answer
1k
views
A result on Lie group actions on 15-dimensional spheres?
In this interview by Eric Weinstein to Roger Penrose, Timestamp 1:24:05., what result is the host talking about?
Transcription of the relevant part:
"If you have two sets of symmetries, known as ...
- 108k
1
vote
1
answer
55
views
Let $G'\triangleleft G<\operatorname{Iso}(M)$ be a normal subgroup. A $G'$-stratum is the union of $G$-strata of lesser dimension
Let $G$ be a group of isometries acting effectively by isometries on a connected Riemannian manifold. And let $G'\triangleleft G$ be a normal subgroup. I am trying to prove that $\dim \operatorname{St}...
- 169
3
votes
0
answers
135
views
Moment map of $\mathrm{O}(n)$-action on $\mathbb{C}^n$
Let $(\mathbb{C}^n, \omega_0)$ be the complex Euclidean space of dimension $n$ with the standard Kähler structure $\omega_0$. I am looking for a Hamiltonian $\mathrm{O}(n)$-action on $(\mathbb{C}^n, \...
- 369
5
votes
1
answer
309
views
Example of closed 4 manifold with $\mathbb{S}^1$ action with 1 fixed point and free away from it
I am looking for a smooth closed 4-manifold $M$ with a distinguished point $x\in M$, endowed with an $\mathbb{S}^1$ action such that the stabilizer of $p\in M\setminus\{x\}$ is trivial and $x$ is ...
- 1,111
1
vote
1
answer
488
views
Homotopy of group actions
Let $G$ be a topological group and $X$ be a topological space.
Let $\alpha$, $\beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous ...
- 16.5k
8
votes
1
answer
360
views
Lie group actions on $S^n$ with some invariant hypersphere but no totally geodesic ones
Does there exist a compact connected Lie group $G$ acting smoothly as isometries on the standard sphere $S^n$ for some $n\ge 3$, so that no totally geodesic hypersphere $S^{n-1}$ is $G$-invariant, but ...
- 28.9k
17
votes
1
answer
505
views
Is a smooth action of a semi-simple Lie group linearizable near a stationary point?
Suppose that $G$ is a semi-simple Lie group that acts smoothly (i.e., $C^\infty$) on a smooth, finite dimensional manifold $M$. Does it follow that the action of $G$ is linearizable near any ...
- 4,713
1
vote
0
answers
72
views
Volume form preserved by the action of $\mathrm{PGL(n+1, \mathbb{R}})$ on $\mathbf{P}^n(\mathbb{C}) \setminus \mathbf{P}^n(\mathbb{R}) $?
I know this is quite an elementary question but I am not an expert in Lie theory.
Does the action of $\mathrm{PGL(n+1, \mathbb{R}})$ on $\mathbf{P}^n(\mathbb{C}) \setminus \mathbf{P}^n(\mathbb{R}) $ ...
- 10.4k
1
vote
0
answers
172
views
sequence definition of proper group action
My understanding is that for an action by a Lie group $G$ on a second countable and Hausdorff differentiable manifold $M$ to be proper, it suffices to show that the map $G \times M \rightarrow M \...
- 101
5
votes
1
answer
255
views
Is the action of $SO(n)$ on the sphere $S^{n-1}$ ballanced?
A subset $B$ of a group $G$ is called balanced if $gBg^{-1}=B$ for all $g\in G$.
An action of a group $G$ on a metric space $X$ is called ballanced if for each non-empty balanced subset $B\subset G$ ...
- 41.9k
2
votes
0
answers
100
views
Effective actions by non-commutative groups have non-commuting fundamental vector fields?
I have a bit of a contradiction in my brain and I was hoping once again that excellent Mathoverflow community could help me out :)
Let $\rho_g$ be the action associated to a non-abelian Lie Group $G$ ...
- 979
3
votes
1
answer
526
views
Orbits of unipotent groups over local fields are closed?
Let $H$ be a connected, unipotent linear algebraic group defined over a local field $k$. Let $H \times_k X \rightarrow X$ be an action of $H$ on an irreducible, affine $k$-variety $X$ which is ...
- 4,189
5
votes
0
answers
210
views
Divisible orientation preserving diffeomorphism which is time-$1$ map of no smooth flow
Is there an orientation preserving smooth diffeomorphism $f$ on a compact manifold $M$ such that
for every $n\in \mathbb{N}$, there is a smooth diffeomorphism $g:M \to M$, as $n$th root of ...
- 356
3
votes
1
answer
460
views
Is there an easy example of group action where the slice theorem produces a non-trivial principal bundle?
Let $\rho$ be a group action by a compact group $G$
\begin{equation}
\rho:G\times M \rightarrow M \\
\rho:(g,p) \rightarrow \rho_g(p)
\end{equation}
Denote the orbit of $p\in M$ by $\...
- 6,546
2
votes
1
answer
123
views
Hamiltonian Group action with infinitely many stabiliser types
What is an example of a connected symplectic manifold $(M,\omega)$, with a Hamiltonian action of $G = U(1) =S^{1}$ with infinitely many stabiliser types?
Infinitely many stabiliser types means that ...
- 5,824
14
votes
1
answer
681
views
If an equivariant map is smooth on diagonal matrices, is it smooth everywhere?
This is a followup from a question I asked on math.SE, which received a helpful answer but unfortunately not a complete one. $\def\Sym{\mathrm{Sym}_{n\times n}}$
$\def\s{\mathrm{Sym}}\def\sp{\s^+}$Let ...
- 15.5k
3
votes
1
answer
116
views
Estimates for radii of slices for proper Lie group actions
Let $G$ be a Lie group acting properly on a smooth manifold $M$, and equip $M$ with a Riemannian manifold that is adapted to the foliation by orbits. The celebrated theorem of Palais is that there ...
- 25.3k
5
votes
1
answer
332
views
Orientability of orbit type strata of Lie group actions
Let $G$ be a compact Lie group that acts on a smooth, finite dimensional, oriented manifold $M$, and suppose that such action preserves orientation, i.e., for each $g\in G$, the diffeomorphism $\mu_g$ ...
- 27.8k