Homogeneous space

It is a set $X$ together with a transitive group action of a group $G$ . There is only one orbit and so $X$ is isomorphic to the space of cosets $G / H$ being $H$ the stabilizer or isotropy group $S t a b_{G} (x)$ for an arbitrary $x \in X$ : the map

φ : G / H \to X,

being $φ (g H) = g h x$ for any $h \in H,$ is bijective.

The choice of $x$ has no influence since the isotropy groups are conjugate.

Basic examples:

$X = R^{2}$ and $G = E (2)$
Any connected topological manifold $X$ is homogeneous with $G = H o m e o m (X)$ , see this answer in Mathstackexchange. The same is true for any connected smooth manifold and its group of diffeomorphisms. But keep an eye: these groups are infinite dimensional.
Any finite set $S$ of cardinal $n$ is homogeneous with $G = S_{n}$ the symmetric group. Indeed, any set is homogeneous with respect to its bijections, I guess.
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I think they are the same as Klein geometrys.

When the isotropy group is the identity, $G \equiv X$ , and $X$ is called a principal homogeneous space or $G$ -torsor. This happens if the group action is also free, as well as transitive, that is, regular.

Proposition. Let $G$ be a connected Lie group and $H$ is a closed subgroup of $G$ such that the homogeneous space $G / H$ is contractible. Then $G / H$ is homeomorphic to a Euclidean space $R^{n}$ for some $n$ .
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Proof. This question in MO $◼$

Homogeneous spaces are used, to my knowledge, in two different ways:

They can be attached to a manifold by means of a solder form, to produce a Cartan geometry.
We can be interested in the submanifolds contained in them, which can be studied by means of extrinsic moving frames.

Intuitive approach

Suppose we have a complicated (real-world) object $X$ which we can describe by means of a well known (mathematical-world) object $S$ . Examples:

A plane piece of land $X$ described by $R^{2}$ .
A list of students $X = {A n t o n i o, C a r l o s, \dots, R o d r i g o}$ described by the set ${1, 2,, \dots, 32}$ .
...

The object $X$ is homogeneous with respect to the group $B i j (S)$ in the sense that:

There is a physical linkage, or external linkage, from $S$ to $X$ , which is introduced externally. Let's call it $ϕ : S \to X$ .
All the possible descriptions of $X$ are given by $P = {ϕ \circ g : g \in B i j (S)}$ .
Suppose we have a distinguished point in $s \in S$ (for example $(0, 0) \in R^{2}$ or $1 \in {1, \dots, 32}$ ). At first it may seem that the point $x = ϕ (s) \in X$ is a privileged point, since it is described by the "main point" of $S$ . But it turns out that

P = ⊔_{y \in X} P_{y}

being $P_{y} = {ϕ \circ t : t \in t_{y} H}$ (with $H$ the isotropy group of $s$ in $B i j (S)$ and $t_{y}$ any map sending $s$ to $ϕ^{- 1} (y)$ ) all the descriptions of $X$ "centered at" $y$ . That is, $P_{y}$ consist of descriptions of $X$ in which $y$ is described by the distinguished point $s \in S$ . It is clear that all $P_{y}$ are "equal", and so we say that $X$ is homogeneous. For example, no point of the land is special, all of them can be chosen to be the center of the world; and no student is special, all of them can be chosen to be associated with the label 1.

Now, it is interesting to now if we can shrink $B i j (S)$ to a subgroup $G$ in such a way that $X$ still conserve this property. Why do we want to restrict? Because there are descriptions which are more important than others. For example, any "crazy" bijection $g \in B i j (R^{2})$ may not give rise to an interesting description $ϕ \circ g : R^{2} \to X$ (maybe it is interesting for an alien, but not for us). We are interested in bijections that do not clutter the set too much (for example, with our "sense" of continuity). The choice of $G$ is again an external data, which depends on the nature of the problem we are interested in. But the group needs to be transitive, in order to conserve homogeneity.

G-descriptions

An element $ϕ \circ g \in P$ is a $G$ -basis or $G$ -description of $X$ in the sense that we can define the coordinates of a point $y \in X$ as

(ϕ \circ g)^{- 1} (y) \in S

The group $G$ not only serves to parameterize the " $G$ -descriptions" of $X$ . It also defines an action on $X$ through $ϕ : S \to X$ . The action of $g \in G$ on $y \in X$ is $g y = ϕ (g (ϕ^{- 1} (y)))$ . If we fix a point $x \in X$ (corresponding to certain $s \in S$ by means of $ϕ$ ) we get a bijection $Ψ : X \to G / H$ (being $H$ the isotropy group of $x$ ) given by

Ψ : y \mapsto g_{y} H

g x \mapsto g H

being $g_{y}$ an element of $G$ carrying $x$ to $y$ . So many times we identify $X$ with $G / H$ , or even start the construction with $G / H$ .

This bijection $Ψ$ let us to express the points of $G / H$ in "coordinates" in $S$ . Given $\bar{g} \in G$ , the element $\bar{g} H$ have coordinates

(ϕ \circ g)^{- 1} (Ψ^{- 1} (\bar{g} H)) = (ϕ \circ g)^{- 1} (\bar{g} x) =

= (ϕ \circ g)^{- 1} (ϕ (\bar{g} (ϕ^{- 1} (x)))) = (g^{- 1} \bar{g}) ϕ^{- 1} (x) =

= (g^{- 1} \bar{g}) s

In particular, if we think in $X = G / H$ described by the set $S = G / H$ by means of the identity and with selected points $s = x = H$ , then an abstract point $\bar{g} H \in G / H$ is expressed in a frame $g \in G$ as

g^{- 1} \bar{g} H

Example
$G = R^{2} ⋊ G L (2)$ , $H = G L (2)$ , $G / H = R^{2}$ , $x = s = τ_{(0, 0)} H$
An element $g = τ \circ A \in G$ is a translation $τ$ following a linear transformation $A$ , and sends the canonical reference system $((0, 0), (1, 0), (0, 1))$ to a new one. The coordinates of a point $y \in R^{2}$ in the new reference system are computed by means of $A^{- 1} τ^{- 1} y$ .
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In particular $G$ itself is a homogeneous space ( $H = 1$ ). Every element $f_{1} \in G$ is, at the same time, a "point", a "movement" and a "frame". Movements and frames are the same, see general covariance and contravariance. Now, given any other point $f_{2} \in G$ , how it is described from the frame $f_{1}$ ? Well, according to this we take

f_{1}^{- 1} f_{2}

What if the action is not transitive?

Well, in this case we can decompose the space in the orbits $S = ⊔_{i} O_{i}$ . The group acts transitively in every $O_{i}$ so we have bijections

Ψ_{i} : O_{i} \to G / H_{i}

where $H_{i}$ is the isotropy group of some point in $O_{i}$ . Then we have several principal bundles $G \to O_{i}$ , not only one. I think this has to do with subrepresentations of groups, but I have to think more on this.