\[ \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\O}{O} \]

The orbits of the action of \(\O(n)\) on \(\mathbb{R}^n\) are \(\{0\}\) and all spheres.

See also Link to heading

• orbits of \(\GL(n)\) on \(\mathbb{R}^n\)
• action of \(\O(n)\) on \(\mathbb{S}^{n-1}\) is transitive