0x67cca0bd March 8, 2025 The orthogonal group \(\O(n)\) acts continuously on \(\mathbb{S}^{n-1}\) via vector-matrix multiplication. Remarks action of \(\O(n)\) on \(\mathbb{S}^{n-1}\) is transitive See also Link to heading action of \(\GL(n)\) on \(\mathbb{R}^n\) action of \(\O(n)\) on \(\mathbb{R}^n\)