Tagged Questions
1
vote
1answer
37 views
Is this distribution involutive?
For two days I've been trying to show the following: Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and consider the smooth distribution $$F=\{F_p=DR_p(e)\mathfrak{h}; p\in G\},$$ where ...
1
vote
1answer
58 views
Question about lie bracket..
Let $G$ be a Lie group with Lie algebras $\mathfrak{g}$ and let $\mathfrak{h}\subseteq \mathfrak{g}$ be a Lie subalgebra. Write $F_p=DR_p(e)\mathfrak{h}$, $p\in G$, where $R_p:G\rightarrow G$ given by ...
2
votes
1answer
57 views
Tangent space at the identity element of a lie group
Let G be a lie group . we know a Lie group is a group with a smooth manifold structure s.t both the multiplication map $m$ and group inversion map $i$ are smooth .
Now by identifying ...
0
votes
0answers
42 views
Definition of g-orbit of a set
Let $g$ be a Lie algebra and $M$ a manifold, what does mean $g$-orbit of $M$?
2
votes
1answer
102 views
Image of Homomorphism of Lie groups
This is exercise from Lee: Introduction to smooth manifolds.
Suppose $f \colon G \to H$ is homomorphism of Lie groups (real, finite-dimensional).
Q: Is image $Im(f) \subseteq H$ a Lie subgroup of H?
...
3
votes
1answer
83 views
Some questions about $S^n$
I have some questions about the $n$-sphere:
I know that for $n=0,1,3$, $S^n$ forms a Lie group and I also know why it's true, but why is it not the case for other $n$?
I have the same question for ...
2
votes
0answers
58 views
Conditions for a group to admit the structure of a Lie group
This question is motivated by a previous one:
Conditions for a smooth manifold to admit the structure of a Lie group
and wants to be a sort of "converse". Here I am taking an abstract group $G$ and ...
4
votes
1answer
71 views
$U(n)/U(n-1)$ as homogeneous space
How can I prove that the quotient $U(n)/U(n-1) \simeq S^{2n+1}$ (where $U(n)$ is the unitary group). Is il correlated with the teory of homogeneous spaces?
0
votes
0answers
68 views
Covering space (Lie groups and their maximal tori)
Let $ G $ be a compact Lie group and $ T $ a maximal torus in $ G $. We define the Weyl group $ W $ as the quotient space $ {N_{G}}(T)/T $, where $ {N_{G}}(T) $ is the normalizer of $ T $ in $ G $. We ...
0
votes
0answers
36 views
Smooth Action of a Finite Group
Suppose $H$ is a finite group acting smoothly on a smooth connected manifold $M$. The action is trivially proper, as $H$ is discrete. If the action of $H$ were also known to be free, i.e. $h\cdot ...
5
votes
1answer
71 views
Isometries from Diffeomorphisms
Given a compact, connected Lie group G of diffeomorphisms on a manifold M, how to construct a Riemannian metric on M such that elements of G are isometries of M?
1
vote
1answer
74 views
Definition of differential of Adjoint representation of Lie Group
Let $g$ be an element of Lie Group $G$, and $\gamma(t) : \mathbb{R} \rightarrow G$ be a path in $G$ such that $\gamma(0) = e$, the identity element of $G$. Denote the tangent space at $e$ as $T_eG$, ...
3
votes
0answers
38 views
Show that the cosets of a closed isotropy group form a manifold
Suppose $G$ is a Lie group acting on the manifold $M$ and $p \in M$ is such that $G_p$, the isotropy group of $p \in M$, is closed in $G$. I'm trying to prove that $G/G_p$ has a manifold structure.
...
1
vote
0answers
36 views
A K-invariant submanifold of G-manifold and fundamental vector fields
Let a (connected) Lie group $G$ act on $M$. Assume that the action is locally free. (In other words, if the fundamental vector field of $X \in \mathrm{Lie(G)}$
$$
\underline{X}(p) := ...
2
votes
2answers
58 views
Is it true that $R^2 = E(2)/U(1)$?
(Just so we're clear: that the Lie group of planar translations $R^2$ is isomorphic to a quotient of the 2D Euclidean Lie group $E(2)$ and the circle group $U(1)$.)
I am trying to prove that $R^2 = ...
3
votes
1answer
95 views
A question on Lie sub-group
Well, definition of Lie subgroup what I know is, a Lie subgroup of a Lie group $G$ is an abstract subgroup $H$ which is an immersed submanifold via the inclusion map so that the group operations on ...
2
votes
1answer
209 views
Two Lie algebras associated to $GL(n,\mathbb{C})$
I have elementary questions about Lie groups and their associated Lie algebras.
Let $G=GL(n,\mathbb{C})$. Then associated to this Lie group is the Lie algebra $M_n(\mathbb{C})$ with the commutator ...
1
vote
1answer
96 views
Conditions for left-invariant one-forms to be closed.
Let $G$ be a connected (semisimple) Lie group with Lie algebra $\frak{g}$. For $\omega \in \frak{g}^*$, we may define a left invariant one-form on $G$ by $\left[ \omega (g)\right] (v)=\omega \left( ...
1
vote
1answer
328 views
Special orthogonal group as a manifold
Knowing that $\mathrm{O}(n,\mathbb{R})$ is a closed submanifold (of the general linear group) and that $\mathrm{SO}(n,\mathbb{R})$ is one of its subgroups with the same dimension, is there a quick way ...
7
votes
2answers
173 views
Do we know this homogeneous space by another name?
Consider the homogeneous space $GL(3)/GL(2) = GL(3,\mathbb{R})/GL(2,\mathbb{R})$ where $GL(2)$ fixes the first coordinate axis (so can be identified with the subgroup of $2\times 2$ blocks sitting in ...
0
votes
0answers
107 views
Transportation of a tangent vector by left-invariant translation in complete Riemannian manifolds
How can i guarantee that a transportation of a vector, defined on the tangent space at a element in complete Riemannian manifold, by left-invariant translation is a parallel transport along geodesic ...
