All Questions
22 questions
2
votes
1
answer
122
views
Slowly increasing smooth mappings with values in a Lie group?
Let $G$ be $l$-dimensional compact Lie group and consider any smooth $F : \mathbb{R}^n \to G$.
Then, the first-order derivative of $F$ at each $x \in \mathbb{R}^n$ can be regarded as a linear mapping $...
- 3,477
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 ...
- 21.8k
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 ...
- 516
1
vote
1
answer
345
views
Is the manifold of complex points of a quotient of compact groups just the tangent bundle?
In great generality a Lie group mod its maximal compact subgroup is contractible (for example this is true for all connected Lie groups). Whenever this is true then the Lie group $ D $ is ...
- 4,081
8
votes
1
answer
599
views
Exact condition for smooth homogeneous to imply Riemannian homogeneous for compact manifolds
Let $ (M,g) $ be a homogeneous Riemannian manifold. That is, the isometry group $ Iso(M,g) $ acts transitively on $ M $. Let $ \pi_1(M) $ be the fundamental group of $ M $. Then $ \pi_1(M) $ has ...
- 4,081
4
votes
1
answer
369
views
A Fréchet space characterization of smooth structures on topological spaces?
For a compact manifold $M$ the space of smooth functions $C^{\infty}(M)$ is a Fréchet space where the seminorms are the suprema of the norms of all partial derivatives. Is there some way to ...
- 185
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
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 $\...
- 639
1
vote
1
answer
186
views
A question regarding the action of a Lie subgroup
Suppose $H$ is a closed subgroup of a Lie group $G$. Then in Lee's book Introduction to Smooth Manifolds (Ch. 9) he showed that the action $H\times G\to G$ $(h,g)\mapsto gh$ is a smooth, free, proper ...
- 1,879
3
votes
1
answer
370
views
Genericity of equivariant embeddings
I'd like to ask an equivariant version of this question.
Let $M$ be a closed manifold equipped with the action of a compact Lie group $G$. By the Mostow-Palais embedding theorem, $M$ can be embedded ...
- 1,903
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
1
answer
152
views
Model geometry uniqueness
Let $ M $ be a compact connected manifold with
$$
M \cong \Gamma \backslash G /H
$$
where $G $ is a Lie group, $ H $ a compact subgroup, $\Gamma $ a discrete subgroup, and $ G/H $ is connected and ...
- 4,081
3
votes
2
answers
2k
views
Is a manifold paracompact? Should it be?
We will say that a Hausdorff topological space $X$ is a smooth manifold if there is an open cover $(U_{\alpha})$ of $X$ and a corresponding collection of homeomorphisms $\varphi_{\alpha} : U_{\alpha} \...
- 1,379
5
votes
2
answers
377
views
Existence of an isotopy in Riemannian manifold
Let $(M,g)$ be a Riemannian manifold, and $p,q\in M$ be two fixed points. We assume $p,q$ are close enough. Say, we assume $p$ and $q$ are in the same normal coordinate chart. It is clear that there ...
- 2,789
9
votes
1
answer
807
views
Formal vector fields vs. (standard) vector fields
Given a smooth manifold $M$, one can consider the Lie algebra $\mathcal{X}(M)$ of vector fields equipped with the standard Lie bracket. This is a standard machinery of differential geometry. Gelfand ...
- 9,330
4
votes
1
answer
236
views
Submanifold of a Lie group whose tangent bundle is invariant under group (left) action
Edit: According to the interesting comment of Tobias Fritz we revise the question.
Assume that $G$ is a Lie group and $M\subseteq G$ is a closed connected smooth submanifold of $G$ containing the ...
- 356
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
3
votes
1
answer
98
views
Locally nilpotent algebraic section of tangent bundle is complete?
Suppose $X$ is a smooth affine algebraic variety over $\mathbb{C}$ and let $V$ be an algebraic vector field (i.e. an algebraic section of the tangent bundle). If $V$ is locally nilpotent, meaning that ...
user75660
2
votes
1
answer
120
views
$E \times_H \mathbb{R}^n$ is isomorphic to the total space of the tautological bundle $\gamma^n$ over $G_n(\mathbb{R}^{n+k})$?
Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & ...
- 85
0
votes
1
answer
1k
views
cartan killing metric [closed]
I know that we can define the killing form on a lie algebra. However, when going to the group manifold, does this give rise to a metric on the manifold? I thought that would be the case, but I cant ...
- 293
17
votes
3
answers
5k
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 ...
- 173
12
votes
3
answers
3k
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).
...
- 2,806