41 votes
Accepted

Rotation Matrix of rotation around a point other than the origin

Your first formula is correct. Remember, the point to which this is applied appears on the RIGHT: $$ T(x,y) * R * T(-x,-y) (P) $$ So to evaluate the expression above, we first translate $P$ by $(-x, -...
  • 88.6k
10 votes
Accepted

What is the interpretation of homogeneous line intersection?

Consider three-dimensional space $\mathbb R^3$. If you normally homogenize by appending $z=1$, that means that your geometry as you know it happens on the $z=1$ plane in space. But it's also possible ...
  • 39k
10 votes

Rotation Matrix of rotation around a point other than the origin

The point is, that you're shifting the coordinate system, not the point. So you don't actually shift the point to the origin, you shift the origin to the point, and then back.
  • 211
9 votes

Riemannian metrics on homogeneous spaces

Let $G$ be a connected Lie group and $H$ a closed subgroup (not necessarily compact). The set $\mathcal{X}_{G,H}$ of $G$-invariant Riemannian metrics on $G/H$ is in canonical bijection with the set ...
  • 16.6k
9 votes

What is the space $\text{SL}(n, \mathbb{C})/\text{SL}(n,\mathbb{R})$?

You can identify $SL(n,\mathbb C)/SL(n,\mathbb R)$ with the set of real structures on the complex vector space $\mathbb C^n$. Here a real structure on a complex vector space $V$ is defined as a map $\...
  • 17.9k
8 votes
Accepted

Are Homogenous countable complete metric spaces always discrete?

Suppose that $X$ is a complete, countable, homogeneous metric space. Clearly $X=\bigcup_{x\in X}\{x\}$ is a countable union of closed sets. Every complete metric space is a Baire space, so the sets $\{...
8 votes
Accepted

An example of a homogeneous, non-symmetric space

Here is my favorite example. Consider the flag space $$G(k,\ell,n) = \{(L,H)\in G(k,n)\times G(\ell,n): L\subset H\}.$$ Here $G(k,n)$ is the Grassmannian of $k$-planes in $n$-space, and $G(\ell,n)$ is ...
7 votes
Accepted

Homogeneous topological space with the fixed-point property

The standard example is the Hilbert cube $[0,1]^\mathbb{N}$. It is homogeneous by a theorem of Keller: see Why is the Hilbert Cube homogeneous? for a discussion. It has the fixed-point property by ...
7 votes
Accepted

The projective space as a homogeneous space

Consider the canonical action of $SO(n+1)$ on $R^{n+1}-\{0\}$. It defines an action of $SO(n+1)$ on $RP^n$ as follows: let $x$ be an element of $R^{n+1}-\{0\}$, denote by $[x]$ the class of $x$ in $RP^...
7 votes

Can a homogeneous space be a manifold with boundary?

Being a boundary point of a manifold can be characterised purely topologically (in terms of local homology groups say). So for each group element $g$ as $x\mapsto gx$ is a homeomorphism of $M$ to ...
7 votes
Accepted

Is it possible to realize $ \mathbb{RP}^2 $ as a linear group orbit?

We can embed $\mathbb{RP}^2$ into the space $M_3$ of $3\times 3$ matrices via the map which takes a line $[v]$ to the orthogonal projection onto $[v]$: $$ [v]\mapsto\frac{vv^T}{\|v\|^2} $$ This gives ...
  • 13.5k
6 votes

Rotation Matrix of rotation around a point other than the origin

These matrices are left-side multiplicated with vector positions, so the order of multiplication is from right to left - on the right side is the first operation, on the left side - the last one.
6 votes
Accepted

Why is $SU(n)/SU(n-1)$ the $2n-1$-sphere?

The group $SU(n)$ has a natural action on $\Bbb C^n$ and also on the unit ball within, the set $\{z\in\Bbb C^n:\|z\|=1\}$ which we can identify with $S^{2n-1}$. Then $SU(n)$ acts transitively on $S^{...
6 votes
Accepted

A non-homogeneous space such that all neighborhood spaces are homeomorphic

Here's a counterexample. Let $X$ be a countably infinite set and choose a partition $$X=\bigcup_{n\in\mathbb{N}} A_n\cup \bigcup_{n\in\mathbb{N}} B_n\cup\bigcup_{n\in\mathbb{N}} C_n$$ where each $A_n,...
6 votes

Homotopy group of the quotient $O(2n)/(U(p) \times U(n-p))$

The fibration $U(p)\times U(n-p) \to O(2n) \to O(2n)/(U(p)\times U(n-p))$ induces a long exact sequence in homotopy groups $$\dots \to \pi_{k+1}(O(2n)/(U(p)\times U(n-p))) \to \pi_k(U(p)\times U(n-p)) ...
6 votes
Accepted

G acts effectively implies $ dim(G) \leq \frac{n(n+1)}{2} $

Let $M=\mathbb{R}$. Then there is an effective action by the $2$-dimensional group of affine maps $\{x\mapsto ax+b|a\in \mathbb{R}\backslash\{0\},\,\,b\in \mathbb{R}\}$. Also another example, where $M$...
  • 10.4k
5 votes
Accepted

Is every Hausdorff homogeneous space also regular?

An example of a homogeneous Hausdorff space which is not regular is obtained by giving $\mathbb R$ the topology where the open sets are those of the form $U \setminus A$ where $U \subseteq \mathbb R$ ...
5 votes
Accepted

1-Forms on $SO(3)$ and $S^2$

Left-invariant vector fields are invariant under left multiplication. However, the flow of a left-invariant vector field acts by right multiplication: For example, if $\theta^i_t$ represents the flow ...
  • 42.2k
5 votes
Accepted

Why are Klein geometries $G/H$?

A Klein geometry is specified by a Lie group $G$ and a closed Lie subgroup $H < G$; sometimes we additionally require that the quotient space $X := G / H$ be connected. In particular, the action $G ...
5 votes
Accepted

Examples of non parallelizable homogeneous spaces $G/H$ where $H$ is discrete

In the case where $H$ is discrete, there exist no such non-parallelizable examples due to the following argument. For $H$ discrete, the map $\pi: G\to G/H$ is a covering map. Let $F: G\to FG$ be a ...
  • 1,090
4 votes

Canonical connection on $CP^n$

The standard action of $U(n+1)$ on $\mathbb C^{n+1}$ preserves the unit sphere $S^{2n+1}$ and commutes with the action of $U(1)$ (multiplication of vectors by complex unit scalars). The quotient by $U(...
  • 3,453
4 votes
Accepted

What is the dimension of the space of planes in $\Bbb R^3$?

Since we already know that the space of directions in $\Bbb R^3$ is $2$ (indeed, it can be identified with the unit sphere $\Bbb S^2 \subset \Bbb R^3$), we can see that the space $\Bbb P^2$ of lines (...
4 votes
Accepted

Metric on Homogeneous Space $G/H$

$\mathfrak{h}$-invariant complements to $\mathfrak{h}\subseteq\mathfrak{g}$ correspond to splittings of the canonical short exact sequence of $\mathfrak{h}$-modules: $$0\to \mathfrak{h}\to\mathfrak{g}\...
4 votes

Riemannian metrics on homogeneous spaces

The answer to your question is no for several reasons. First, just because you can write $M = G/H$ does not mean that any of the natural metrics on $M$ have isometry group $G$ - it is sometimes ...
  • 46.7k
4 votes
Accepted

Spheres as Homogeneous Spaces

The first part of the question has a positive answer, since we have $$ S^n\cong SO(n+1)/SO(n) $$ also for odd $n$. For the case $SU(n)$ this is not possible in this way, however it might be realized ...
4 votes
Accepted

Let $G$ be a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogeneous space $\frac{G}{H}$ are connected, then $G$ is connected.

Your argument has a significant issue in the paragraph after the theorem you quote: $H$ almost never acts transitively on $G/H$ by any action, and the induced action of $H$ on $G/H$ coming from the ...
  • 46.7k
4 votes
Accepted

Finding the orbits of $SL(n+1)$ and a parabolic subgroup $P$ in a representation?

Even for reductive Lie groups $G$, the general problem of determining the $G$-orbit decomposition of a linear representation of $G$ is very difficult---or even “hopeless”, see this MathOverflow post. (...
4 votes
Accepted

Finding explicit elements in $\text{SL}(3,\mathbb{R})/\text{SO}(3)$ whose simultaneous stabilizer in $\text{SL}(3,\mathbb{R})$ is the identity

You can (equivariantly) identify the quotient $SL(3,R)/SO(3)$ with the space of positive-definite quadratic forms in 3 variables with determinant (of the associated symmetric matrix) equal to $1$. The ...
  • 85.4k
4 votes
Accepted

What is the difference between an homogeneous and an isotropic space

Speaking of a connected Riemannian manifold $(M, g)$, the structure is Homogeneous if, for every pair of points $p$ and $q$, there exists an isometry $f:M \to M$ such that $f(p) = q$. (The ...

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