0
votes
0answers
20 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
1answer
27 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
46 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
55 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
0answers
52 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
24 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
83 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
91 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
59 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
25 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
47 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
96 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
64 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
50 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
40 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
33 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
41 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
39 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 ...
0
votes
1answer
25 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
45 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
82 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
39 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
89 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
37 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
62 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
85 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
29 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
40 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
89 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
82 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
49 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 ...
0
votes
1answer
51 views

What is this dimension?

What is the real dimension of the cone of $2$ by $2$ Hermitain matrices with at lease one eigenvalue that is $0$?
3
votes
2answers
60 views

Riemannian Manifolds with $n(n+1)/2$ dimensional symmetry group

Given a $n$-dimensional connected Riemannian manifold $(M,g)$, its symmetry group $G$ can be considered as a subbundle of orthonormal frame bundle of $M$ (which I call $F_OM$), yielding: $$\dim G\le ...
1
vote
1answer
59 views

Lie algebra homomorphism and action on a manifold

In Introduction to smooth manifolds Lee says on page 527: If $\mathfrak{g}$ is an arbitrary finite-dimensional Lie algebra, any Lie algebra homomorphism ...
2
votes
0answers
34 views

Analytic/Smooth/Continuous maps between a manifold and itself

Let us suppose that $M_{\omega}$ is a connected real-analytic manifold of dimension $n$. Then there is an associated smooth structure, $\mathcal{C}^r$ structure ($r$ non-negative integer) on it. Let ...
3
votes
0answers
88 views

Vector fields on smooth manifolds and Lie algebras

I'm currently studying differential geometry on smooth manifolds using differential forms and I'm trying to apply it to what I have learned earlier about Lie groups, but something doesn't seem to ...
4
votes
0answers
103 views

Is $G$ a lie group if left multiplication is smooth and multiplication is smooth near $e$?

Suppose $G$ is a smooth manifold and also a topological group. Also suppose that left multiplication $L_g : G \rightarrow G$ is smooth for any $g \in G$. Finally suppose that the multiplication map is ...
2
votes
0answers
47 views

Group generated by several vector fields

I have two (or more) smooth and integrable vector fields $v,w$ on a smooth manifold $M$. Each generates a flow map $\Phi_v$,$\Phi_w$ that forms a single parameter Lie group of diffeomorphisms. Let's ...
0
votes
0answers
17 views

Continuous groups of Transformations [Reference request]

I am considering reading the book : 'Continuous Groups of Transformations' by Luther Pfahler Eisenhart. It seems to have a very interesting table of contents. However this is quite old and I am ...
2
votes
1answer
318 views

Studying Navier-Stokes equations using differential geometry

I study Navier-Stokes in $\mathbb{R}$. But I am interested in applying Differential Geometry for these equations. If I extend my domain to a torus, would this enable me to use DG?
2
votes
2answers
33 views

Can I identify $S_k(V)$ with an homogeneous space?

I'm in trouble with a question: Let $V$ be an $n$-dimensional vector space over $\mathbb R$. Can I identify the manifold, $$S_k(V):=\{(X_1, \ldots, X_k): X_1, \ldots, X_k\in V\ \textrm{are linearly ...
2
votes
0answers
60 views

The pushforward under the left action in the group of units of a Clifford algebra

The following I know to be true: let $A$ and $B$ be elements of $GL(m,\mathbb{R})$ and let $X \in T_BGl(m, \mathbb{R})$ and let $L_A:Gl(m, \mathbb{R}) \to GL(m, \mathbb{R})$ be the left multiplication ...
2
votes
1answer
56 views

Fundamental group of a component of $GL_n({\bf R})$

Let $G$ be a component of $GL_n({\bf R})$ such that element has a positive determenant. (1) Since it contains $SO(n)$, $\pi_1(SO(n))$ ? What is a fundamental group of $G$ ? (2) It has a curvature ...
1
vote
0answers
23 views

$S^1$-curves on a Lie group $G$ under additive and multiplicative notation.

I have been trying to do computations for objects of the based loop group and have been embarrassingly frustrated by the following: Let $G$ be a compact, connected, simply connected Lie group with ...
1
vote
0answers
106 views

Construction of line bundles on the flag variety

Edit: The following is phrased in terms of algebraic geometry, but can be thought of analytically as well. Hence I added some tags... I am a bit confused about the subject in the title. For ...
6
votes
0answers
93 views

Translating a passage of a paper by L. Bérard Bergery

I am currently studying the following paper on Einstein manifolds: L. Bérard Bergery, Sur de nouvelles variétés riemanniennes d'Einstein, Inst. Elie Cartan, Univ. Nancy №6, 1-60 (1983). I have ...
1
vote
1answer
50 views

an identity related to moment map

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra, and $X,Y\in \mathfrak{g}$ and also let $\mu:M\to \mathfrak{g^*}$ be moment map($M$ smooth manifold) then prove the following equality ...
1
vote
0answers
39 views

A question about coadjoint orbit

If the coadjoint orbit $\Omega\subset \mathfrak{g^*}$ be contractible then prove that $\Omega$ is integral , i.e., $\int_C \omega\in \mathbb{Z}$ for every integral singular 2-cycle $C$ in $\Omega$, ...