Visualization of Lie groups

Proposition
Given a group action $G \times M \mapsto M$ free and transitive, fixed $m \in M$ we have a bijection

G \overset{≅}{⟼} M

with $g \mapsto g m$
$◼$
See principal homogeneous spaces in homogeneous space .
This result can be proven easily and let us use the manifold $M$ to visualize $G$ .

If the action is not free (but still transitive), we cannot visualize $G$ (for example, $S O (3)$ acting over $S^{2}$ ). Anyway, observe first that the stabilizer of every $x \in M$ are all conjugate of each other. And second, if we fix $x \in M$ , we can establish a surjection $π_{x} : G \mapsto M$ , $π_{x} (g) = g x$ , that gives rise to a bijection

{\bar{π}}_{x} : G / G_{x} \to X

From here we have a natural bijection

G ⟶ ⨆_{m \in M} {m} \times π_{x}^{- 1} (m)

that could be translated to a bijection

G ⟶ \tilde{M} = ⨆_{m \in M} {m} \times G_{m}

if we had a "preferred choice" of a $t_{m} \in G$ such that $t_{m} x = m$ for every $m \in M$ (need to be proven but I think is easy). Take $g \mapsto (g x, g t_{g x}^{- 1})$ ...
In this circumstances, we can even define an action of $G$ at $\tilde{M}$ :

(g, (m, h)) ⟼ (g m, g h g^{- 1})

But it is not necessarily transitive or free.

But on the other hand, since the action of $G$ on itself is free and transitive, we can copy this action to $\tilde{M}$ with the previous bijection and obtain a new action on $\tilde{M}$

(g, (m, h)) ⟼ (g m, g h t_{m} t_{g m}^{- 1})

For example, we can see $S O (3)$ as the sphere bundle of $S^{2}$ . Remember

S M := {(x, v) : x \in M, v \in T_{x} M, | v | = 1} =

= ⨆_{x \in M} {x} \times S^{1}

This is this way because if we fix a point $P$ in $S^{2}$ and define $π_{P} : S O (3) \mapsto S^{2}$ taking the action on $P$ , we have the bijection of $S O (3)$ with $⨆ {x} \times π_{P}^{- 1} (x)$ and, even more, we have preferred transformations from one point to another: the great circle paths "with angle lesser or equal to $π$ ". So (following result above) we have

S O (3) \overset{\equiv}{⟼} ⨆_{x \in S^{2}} {x} \times S O (2)

given explicitly by

g ⟼ (g P, g t_{g P}^{- 1})

where $t_{Q}$ is the great circle path from $P$ to $Q$ .
This bijection is, indeed, a diffeomorphism, as can be proven with the map

Φ : S (S^{2}) \to S O (3)

Φ (x, v) := [x, v, x \times v]

that sends the pair of vectors in $R^{3}$ $(x, v)$ to the orthogonal matrix $[x, v, x \times v]$ .

But let's return to our original map and take $P = (1, 0, 0)$ and think of several examples to "see" the elements of $S O (3)$ .

The identity $i d \in S O (3)$ corresponds to $P$ itself with the identity in $S O (2)$ .
Rotation of axis $x$ and angle $θ$ , $R_{x}^{θ} \in S O (3)$ , corresponds to $(P, R^{θ})$ .
Rotation of axis $z$ and angle $θ$ , $R_{z}^{θ} \in S O (3)$ , corresponds to $(R_{z}^{θ} P, i d)$ .
Rotation of axis $y$ and angle $θ$ , $R_{y}^{θ} \in S O (3)$ , corresponds to $(R_{y}^{θ} P, i d)$ .

In general, rotation $R_{e}^{θ}$ through an axis no orthogonal to $P$ yields $(Q, R^{α})$ where $Q = R_{e}^{θ} P$ and $R^{α} = R_{e}^{θ} t_{Q}^{- 1} \in S O (2)$ , with $α$ an angle that measures how much $P$ deviates from the great circle corresponding to the axis $e$ .
Hypothesis: a point $(Q, R^{α}) \in ⨆_{x \in S^{2}} {x} \times S O (2)$ comes from the composition of the rotation $t_{Q} \in S O (3)$ with a rotation of angle $α$ an axis $Q$ itself...

To sum up, we would have something like this: fixed $P \in S^{2}$ , rotations around great circles passing through $P$ and angle $θ$ are encoded in the arrival point of $P$ , $Q$ . If the rotation is not along a great circle, is encoded in the same final point $Q$ but with an "internal rotation" of angle $γ$ (i.e., an element of the $S O (2)$ of $Q$ ) related to the angular displacement $α$ of the axis of the rotation respect to the ideal great circle rotation taking $P$ to $Q$ with angle lesser than $π$ . By mean of examples I guess that $γ = 2 α$ :
rotations3.jpg|400
Possible rotations other than the great circle of axis $e_{P Q}$ taking $P$ to $Q$ arise when we choose a different axis $e$ along the "perpendicular bisector" of $P Q$ . The extremal case is when $e = - e_{P Q}$ ( $α = π$ ), and then they begin to be repeated (although in a complementary way).
Keep an eye: this is not diffeomporhic to the 3-sphere $S^{3}$ . $S^{3}$ is diffeomorphic to the covering group $\tilde{S O (3)} = S U (2)$ , the double cover of $S O (3)$ . See unitary matrix#Unitary matrix#What is SU 2 topologically.