Example: the sphere as a homogeneous space

Consider the standard action of $SO(n{+}1)$ on $\mathbb R^{n+1}$, hence on the unit sphere

This is a smooth Lie group action .

Transitivity and stabilizer (explicit)

Fix the “north pole” $p=e_{n+1}=(0,\dots,0,1)$. For any $x\in S^n$, there exists $g\in SO(n{+}1)$ with $g\cdot p=x$ (choose an orthonormal basis sending $e_{n+1}$ to $x$), so the action is transitive .

The stabilizer of $p$ consists of rotations preserving the orthogonal complement $p^\perp\cong\mathbb R^n$, hence

as an embedded Lie subgroup (see stabilizer ).

Identification with a coset space

Define

This is well-defined because elements of $SO(n)$ fix $p$. It is bijective by transitivity and the stabilizer description, and it is a diffeomorphism once we use the standard smooth structure on the coset space $SO(n{+}1)/SO(n)$ (which exists because $SO(n)$ is closed; compare the Closed Subgroup Theorem ). Thus

as a homogeneous space .

Dimension check (concrete calculation)

Using $\dim SO(m)=\frac{m(m-1)}{2}$,

consistent with the identification.

Special case. For $n=2$, this reads $S^2\cong SO(3)/SO(2)$, connecting directly to the $SO(3)$ example .