3
votes
1answer
34 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
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 ...
7
votes
1answer
75 views

Is every Lie group the automorphism group of a riemannian manifold?

Given a finite-dimensional Lie Group $G$, is there always a Riemannian manifold $M$, such that $G$ is the group of isometries of $M$?
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
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 ...
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 ...
0
votes
0answers
21 views

Abelian Lie Group [duplicate]

Take a Lie group G and consider the tangent space at one of its points. In particular, the tangent space at the identity e is usually denoted by g := $T_e G$. Can you prove that, if G is an abelian ...
0
votes
0answers
76 views

About the riemannian metric on the torus

Iwas ask if the torus admits a $\mathbb{R}$-invariant riemannian metric, I think in use the fact that the torus is homeophorphic to $\mathbb{R}^2/\mathbb{Z}^2$ and use the action of $\mathbb{R}^2$ on ...
0
votes
0answers
35 views

Bi invariant Riemannian metric on a Lie Group

I'm trying to find an example when G is a lie group that admits a bi invariant riemannian metric and H a is closed subgroup wich the manifold $G/H$ does not admit a $G$-invariant riemannian metric. ...
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
1answer
33 views

Exponential map on $SO(3)$

(1) As I read some article in here ( I cannot found ), so we know that $$ {\rm exp} \ (T_eSO(3)) \neq SO(3) $$ ( ${\rm diag}(-1,-1,1)$ cannot be covered by ${\rm exp}$ ) But there exists some open ...
2
votes
0answers
40 views

O'Neill Formula in terms of Exterior Derivative of Killing Form

O'Neill Formula : Consider a fibration $\pi : (M ,g)\rightarrow M/G$ where $G$ has only one orbit type. Then we have $$ K_{M/G} (d\pi V, d\pi W) = K_M(V,W) + \frac{3}{4} | [V,W]^V |^2_g$$ where $V,\ ...
1
vote
0answers
95 views

Metric Tensor on Lie Group for Left Invariant Metric

Let $G$ be a Lie group and $Q$ be a biinvariant metric. If $h$ is any positive definite scalar product on $T_eG$ then we have a left invariant metric $h$ on $G$ : $$ h_g (dL_g X, dL_g Y) = ...
1
vote
1answer
49 views

Addition of Fundamental Vector Fields

If we define a fundamental vector field, i.e., $$ X^\ast =\frac{d}{dt}|_0 \exp(tX)\cdot p $$ where $p\in M=G/K$, Question 1 : then for $X,\ Y\in (T_eK)^\perp$, we have $$ X^\ast + Y^\ast = ...
0
votes
1answer
46 views

Way distinguishing whether or not complex manifold

$SU(3)$ has dimension 8. Why is this not a complex manifold ? Thank you in advance.
3
votes
1answer
132 views

What is a principal orbit

I am currently reading a paper about Einstein manifolds. There is a comment where I don't know exactly the meaning of the words, namely a certain metric has a group of isometries of dimension $4$ ...
3
votes
1answer
75 views

Dimension of isometry group of complete connected Riemannian manifold

Given an $n$-dimensional geodesically complete connected Riemannian manifold $M$, we want to prove that the dimension of its isometry group is $$\dim {\rm ISO}(M) \leq \frac{n(n+1)}2.$$ Does it ...
-3
votes
1answer
114 views

Existence of a left-invariant $n$-form on a Lie group of dimension $n$

This Do Carmo, Riemannian Geometry, Chapter 1, Exercise 7: Show that there exists a left invariant differential $n$-form $\omega$ on $G$ ($G$ is a compact connected lie group and $\dim G=n$). ...
1
vote
0answers
59 views

Asymptotic invariants of infinite groups

I am reading a Gromov's book " Metric Structures for Riemannian and Non-Riemannian Spaces ". Consider the following concept : $$ distort(X)\doteq sup \frac{length\ dist|_X}{dist|_X} $$ That is for ...
8
votes
0answers
67 views

Are there more embeddings $U(2) \hookrightarrow SO(4)$?

It is easy to prove that $SO(4)$ acts transitively and freely on $S^2$ with fiber $U(2)$. Therefore, we can identify each point of $S^2$ with a particular embedding $U(2) \hookrightarrow SO(4)$. My ...
4
votes
1answer
85 views

Does the $O(n)$ bundle of a manifold depend on the metric?

Let $g_1$ and $g_2$ be two Riemannian metrics on a manifold $M$. These induce two $O(n)$ bundles on $M$, whose fibers over each point $x\in M$ are the groups of orthogonal transformations of $T_x M$ ...
0
votes
0answers
41 views

Detail in polar action

I am reading a paper "Tits geometry and positive curvature - Fang, Grove, and Thorbergsson" See the following site http://arxiv.org/pdf/1205.6222.pdf In page 7, the 9-th line from the bottom ...
5
votes
1answer
135 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?
3
votes
2answers
103 views

SO(5)-invariant metrics on the 4-sphere

Are there any examples of Riemannian metrics on $S^{4} \subset \mathbb{R}^{5}$ that are not SO(5)-invariant? Or are all metrics on the 4-sphere SO(5)-invariant? Hope my question is not too trivial ...
6
votes
1answer
204 views

How to identify $SL(2,\mathbb{C})/SU(2)$ and the hyperbolic 3-space?

I know that every coset representative $g\in SL(2,\mathbb{C})$ for $SL(2,\mathbb{C})/SU(2)$ can be chosen of the form $$ g = \left( \begin{array}{cc} \sqrt{t} & \frac{z}{\sqrt{t}}\\ 0 & ...
3
votes
0answers
132 views

Isometry group of a Lie group

I'm having trouble dealing with the following question : what is the isomety group of $\mathbf{PSL}_2(\mathbb{R})$ viewed as a Lie group with its Killing form ? For the record, its Killing form is the ...
3
votes
1answer
228 views

A question about left invariant vector fields

Let $G$ be a Lie group with bi-invariant metric $\langle , \rangle$ and $X,Y,Z$ left invariant vector fields in $G$, how to conclude that $X\langle Y,Z\rangle=0$?
2
votes
1answer
65 views

$\operatorname{Isom}{(M)}$ has Lie-structure for M metrizable manifold

Suppose $M$ is a smooth and metrizable manifold. Then $\operatorname{Isom}{(M)}$ can be given the structure of a Lie group, so that the action of $\operatorname{Isom}{(M)}$ on $M$ is still smooth. I ...
3
votes
1answer
382 views

Levi-Civita connection of a left-invariant metric

How do I compute Levi-Civita connection of a left-invariant metric on a Lie group in a neighbourhood of $1$ by knowing only its Lie algebra and the metric form on it? I know it's possible because a ...
1
vote
0answers
158 views

Are Lie Groups Homogeneous Spaces?

Is any Lie Group a homogeneous space?