1
vote
0answers
37 views

Finding a lie group structure on $\mathbb R^n\setminus\{0\}$

I want to find all maps $g: \mathbb R^n\setminus \{0\} \rightarrow GL_n(\mathbb R)$ which satisfy the properties $g$ is differentiable and injective $g(g(a)b) = g(a)g(b)$ for all $a,b\in\mathbb ...
1
vote
1answer
71 views

What makes a Lie Group a Differentiable Manifold?

I've recently been trying to glance at the definition of a Lie group, but I'm not clear as to why a Lie group is defined the way it is, and why this becomes a smooth manifold. For example, if we have ...
1
vote
0answers
22 views

Why does the exceptional Lie group $G_2$ have dimension 14?

In ''Compact manifolds with special holonomy" by D. Joyce, on p. 242, the group $G_2$ is defined to be the subgroup of $GL(7,R)$ preserving the $3$-form: $$ \varphi_0 := dx_{123} + dx_{145} + ...
3
votes
0answers
67 views

Affine connection on a Lie group.

Let $G$ be a Lie group. For $g \in G$, we can define a diffeomorphism $l_g: G \to G$ by $l_g(x)=gx$, and a bundle map ${l_g}_*:TG \to TG$. Then, I guess that we can obtain the affine connection on $G$ ...
4
votes
1answer
103 views

A question on the unit tangent bundle of the sphere and $SO(3)$

Let the unit tangent bundle be defined as follows: $$T^1S^2=\{(p,v)\in \mathbb R^3 \times \mathbb R^3 | |p|=|v|=1 \text{ and } p \bot v \}$$ Let $SO(3)$ be the group of rotations of $\mathbb R^3$. ...
7
votes
1answer
148 views

Special conformal killing fields - solving for integral curves.

For each $b\in\mathbb R^d$, let a vector field $X_b:\mathbb R^d\to\mathbb R^d$ be defined as follows: \begin{align} X_b(x) = 2(b\cdot x)x - x^2 b, \end{align} where $x^2 = x\cdot x$. This is the ...
1
vote
1answer
46 views

Is this a tangent bundle and what is the meaning of this exercise

I intend to solve the following exercise but I would like to have some help with understanding the ''big picture'': Exercise. Describe a natural 1 to 1 correspondence between elements of $SO(3)$ ...
1
vote
1answer
41 views

Left-invariant Riemannian metric on $SO(3)$

Let's consier the manifold $SO(3)$. First problem is to show that $T_I SO(3)$ is a space of skew-symmetric matrices $3\times 3$. How can I deduce it? Then I have to prove there exists exactly one ...
1
vote
1answer
84 views

What does it mean to be a real Lie group

What does it mean to be a real Lie-group ? For example it is said that $SU(N)$ is a real Lie-group. While for example for $SU(2)$ the 2 dimensional matrix-representation consists of the Pauli ...
0
votes
0answers
18 views

If $M$ has hyper-kaehler structure then $M//G$ has hyper-kaehler > structure?

A) Let $M$ is a non-compact manifold and $G$ be a compact Lie group which acts on $M$ and preserves complex structure then If $M$ has Kaehler manifold, then the symplectic quotient of $M$, i.e, ...
4
votes
0answers
67 views

When do invariant measures arise from smooth differntial forms?

It is well known that the Haar measure of a Lie group $ G $ arises from a invariant differential form density $ |\omega| $ (of top dimension). Also, we know that if we have a closed subgroup $ H \leq ...
4
votes
1answer
59 views

The diffential of commutator map in a Lie group

Leb $G$ be a Lie group and $f:G\times G\rightarrow G$ be the commutator map $:(x,y)\mapsto xyx^{-1}y^{-1}$. How to obtain the Lie bracket in the associated Lie algebra of $G$ from the derivatives of ...
-2
votes
1answer
39 views

reduced space of coadjoint orbit

Let $G$ be a compact Lie group and $\lambda\in \frak{g}^*$$=(Lie G)^*$ and $O_\lambda$ be the Coadjoint orbit through $\lambda\in \frak{g}^*$ and $\mu:O_\lambda\to\frak{g}^*$ be the moment map, ...
2
votes
0answers
56 views

Gradient of a real-valued function on SO(3)

I have struggling with a problem of evaluating the gradient of a cost function on the Lie group of rotations: SO(3). The cost is the following: \begin{equation} ...
3
votes
0answers
47 views

Actions of Weyl group

I get a feeling what I am going to ask is very standard and classic, but I am not able to find any reference. Any answer or reference would be appreciated. Let us assume that $G$ is a simply ...
3
votes
1answer
55 views

tangent/normal space to set of symmetric isospectral matrices

Let $\Lambda = \{\lambda_1, \ldots, \lambda_n\}$ be a set of $n$ distinct real numbers. $M_n(\mathbb{R})$ denotes the set of all $n \times n$ real matrices, and for $B\in M_n(\mathbb{R})$, $B^T$ ...
0
votes
0answers
25 views

Global section $$s:T^*G\to Sp(T^*G)$$

Let $G$ be a Lie group and then how can we define the global section $$s:T^*G\to Sp(T^*G)$$ Where $Sp(T^*G)$ means symplectic frame bundle of $T^*G$.
0
votes
0answers
65 views

How can we extend the proof that Lie groups are orientable to parallelizable manifolds?

Let $G$ be a Lie group with identity element $e$ and dimension $n$. I know we can take a non-vanishing $n$-form (call it $\omega$) on the vector space $T_eG$. For a Lie group, as we have an ...
1
vote
2answers
48 views

Lie algabra of R^n

Until now the only example of lie groups I have seen are subgroups of $GL_n$. Today I had the idea, that also $G=(\mathbb R^n,+)$ must be a lie group ($(\mathbb R^n,+)$ is a group with the ...
3
votes
1answer
57 views

The Lie derivative by an infinitesimal action is invertible at an isolated zero point

Let a compact Lie group $G$ act on a manifold $M$. Fix $X \in \mathfrak g$ and we write $X_M$ for the infinitesimal action of $X$. Assume that $p \in M$ is a zero point of $X_M$. Define a linear map ...
1
vote
2answers
71 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 ...
3
votes
1answer
97 views

What is the kernel of a Maurer-Cartan form?

The Maurer-Cartan form on the Lie group $Gl(n,\mathbb{R})$ is a one-form taking values in $\mathfrak{gl}(n,\mathbb{R})$ as defined in the link. It has a rather concrete "extrinsic definition" as ...
1
vote
0answers
27 views

Volume form for $SE(n)$ and/or $E(n)$. [duplicate]

I wonder what happens when you construct the Tiling spaces considering the natural action of $SE(n)$ or $E(n)$ rather than $\mathbb R^n$. In order to do that, I need to understand both the riemannian ...
2
votes
2answers
88 views

Proving a submanifold of $SL_2(\mathbb{R})$

I already showed that $SL_2(\mathbb{R})$ is a 3-dimensional manifold. Now I want to show that the subspace $E$ of symmetric matrices whose eigenvalues are positive in $ SL_2( \mathbb{R})$ is a ...
6
votes
1answer
93 views

How to obtain a diffeomorphism between $\mathrm{SL}(2,\mathbb{R})$ and $ (\mathbb{C}\!\smallsetminus\!\{0\})\times\mathbb R$?

Could someone please give me a tip on how to show that the map $\mathrm{SL}(2,\mathbb{R}) \to (\mathbb{C}\setminus\{0\})\times\mathbb{R}$ \begin{equation}\begin{pmatrix} a & b\\ c & d ...
0
votes
0answers
63 views

Proof of Horn theorem with moment map

Please look at this problem: Let $\mathcal{H}$ be the space of $(n,n)$ hermitian matrix. $\phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*}$ ...
2
votes
0answers
29 views

Existence of maximal tori in infinite-dimensional Lie groups

Reading about Lie groups and maximal tori I came up with a lemma that states that any Lie group $G$ has maximal tori. The proof goes like this: firstly, it is proven that if $H \subset G$ is a proper ...
0
votes
1answer
41 views

Question related to tangent space of $U(n)$ at a matrix $g\in U(n)$

I was working on a homework problem that involved showing that the map $f:U(n)\rightarrow S^1,g\mapsto det(g)$ is a submersion (which is given here) And the following question emerged: Given $g\in ...
0
votes
1answer
64 views

Tangent space of $\mathfrak{ so}(3)$ Lie algebra

Very basic question and the terminology makes it difficult to find a reference. I just know the basics of differential geometry but my question is simple. Is the tangent space at the point ...
2
votes
2answers
101 views

Unitary group and unit circle

Let $U(n)$ denote the group of complex unitary matrices, let $S^1$ be the unit circle in the complex plane. Then the map $$f:U(n)\to S^1,\quad f(A)=det(A)$$ is a group homomorphism and a submersion. ...
0
votes
1answer
69 views

A question about differential forms on Lie groups

Let $G$ is a Lie group and $\mathfrak{g}$ is its Lie algebra and $\mathfrak{g}^{\mathbb{C}}$ be its complexification. Also assume that $\mathfrak{h}\subset \mathfrak{g}^{\mathbb{C}}$ be its ...
0
votes
0answers
54 views

Isometries of a general metric

For a general (pseudo-)Riemannian manifold, i.e. in which the interval $ds$ can be written $ds^2 = g_{ab}\,dx^a \,dx^b$, is there a general prescription for finding the group of isometries- by ...
1
vote
0answers
53 views

Any material on complexification?

These days, I met a problem on linear algebra: Suppose $A,B$ are real matrices. If there's a complex unitary matrix $U$ such that $U^*AU=B$, where $U^*=\overline U^\top$, namely, the conjugate ...
1
vote
1answer
42 views

How is the action of a Lie group element on a tangent vector defined?

I am trying to understand the concept of a left-invariant vector field, much as in this question here. I am not clear on what is meant by "derivative of left-multiplication by $g$". How is this ...
1
vote
0answers
53 views

Source for Differential Manifolds/Geometry Questions?

I'm looking for a good source for a large collection of Differential Manifolds/Geometry questions covering a subset of the following topics: inverse function theorem, local coordinates, induced ...
6
votes
0answers
43 views

Two definitions of real flag manifolds: do they coincide?

Let $G$ be a real semisimple Lie group with finite center. Definition 1 A real flag manifold is a homogeneous space $G/Q$ where $Q$ is a parabolic subgroup of $G$. Definition 2 Let $K$ be a ...
1
vote
1answer
34 views

Flag manifolds (=R-spaces): quotients by parabolic subgroups vs. isotropy representation

Real flag manifolds (also known as R-spaces) can be defined in two ways which I believe are equivalent although some fine print may have escaped me: as a quotient of a semisimple real Lie group $G$ ...
1
vote
0answers
20 views

Irreducible Decompositions of the Space of Sections of an Equivariant Vector Bundle

Let $G$ be a compact semi-simple Lie group, and $X = G/H$ a homogeneous space of $G$. If $V$ is a vector bundle over $X$, then is it true that $\Gamma^\infty(V)$ always has a decomposition into finite ...
0
votes
1answer
58 views

Coordinate expression of left invariant vector fields on $SU(1,1)$

I'm trying to work out some differential geometry of the matrix Lie group $SU(1,1)$. It is the group of $2\times 2$ complex matrices such that: $$ (U^{*})^{T}\eta U=\eta $$ where: $$ ...
3
votes
0answers
84 views

How was this Lie algebra found?

In a paper the author lists, without justification, generators for a Lie algebra. I would be grateful if someone could justify these choices and perhaps suggest how I might have found them for myself. ...
0
votes
1answer
43 views

Euler characteristic of 2-dimensional compact Lie Groups

I'd like to know why the Euler characteristic of $G$, a compact Lie Group of dimension 2, is zero. I'm aware of the fact that this is true not only for dimension 2. The point is that I'm not familiar ...
1
vote
2answers
138 views

A left invariant vector field on a Lie group

Let $G$ be a matrix Lie group. Let $v$ be a left invariant vector field on $G$ and $v_1 \in \frak g$, where $\frak g$ is a Lie group of $G$. Let $v_1$ be its value at the identity. We define $\phi_t ...
2
votes
1answer
82 views

$X\in \mathfrak{g}$ means flow commutes with left-translation

Suppose $X\in \mathfrak{g}$ is a left invariant vector field on a Lie group G. In this article it mentions that The fact that our vector fields satisfy $L^*_gX = X$ implies that the flow ...
2
votes
2answers
63 views

Characterization of differentiability via Lie derivatives

Yesterday I asked this question in MathOverflow but did not receive an answer yet. I want to try my chance here too, since I am in kind of a hurry. Answers will be much appreciated. I intend to ...
2
votes
1answer
99 views

How to show that $ G $ is a Lie group?

Problem: Let $ G $ be a group that is also a smooth manifold, and suppose that the mapping $ (x,y) \mapsto x y $ is smooth. How can we show that $ G $ is a Lie group? This is a problem from ...
0
votes
1answer
30 views

Matrix multiplication in $SO(3)$ that fixes row

I want to find all matrices $G \in SO(3)$ that do not change the first row of elements in $SO(3)$ when right multiplying by $G$, i.e. $$ \{ G \in SO(3): \forall A \in SO(3) \quad A = \begin{pmatrix} ...
0
votes
1answer
54 views

Does local flow of left-invariant vector field commute with the left-translation operator?

Let $G$ be a Lie group and $X$ a left-invariant vector field over $G$ (i.e. $\forall g,p\in G: (D_p l_g)(X_p) = X_{gp}$ whereby $l_g$ is the map $G\rightarrow G:p\mapsto gp$). Let $\phi_t$ be the ...
1
vote
2answers
101 views

Vector field invariant under transitive action: restricts to free transitive action?

Thinking about how you can put vector fields on homogeneous spaces that respect the homogeneity, I'm interested in the following situation: Let $V$ be a nonzero vector field on a manifold $M$, let ...
3
votes
3answers
157 views

What is the dimension of this Grassmannian?

Why is $2\times 3$ the dimension of $Gr_2(\mathbb{R}^5)$? and can one use the dimensions of Lie groups to derive this dimension? Note: $Gr_2(\mathbb{R}^5)$ denotes the Grassmannian of all ...
1
vote
1answer
51 views

A slice orthogonal to each orbit

Assume that a compact (connected) Lie group $G$ acts on a manifold $M$. We choose a $G$-invariant Riemannian metric on $M$ and a point $p \in M$. Then using the exponential map at $p$, we can obtain a ...