The \(n\)-dimensional quaternionic projective space \(\mathbb{H}\mathbb{P}^n\) consists of all one-dimensional quaternionic subspaces of \(\mathbb{H}^{n+1}\). It is a quotient space induced by \(q\colon \mathbb{H}^{n+1}\setminus \{0\}\to \mathbb{H}\mathbb{P}^n\).
Remarks
- The quaternionic projective space \( \mathbb{HP}^n \), given by \[ \mathbb{HP}^n := \{ \text{1-dimensional subspaces of } \mathbb{H}^{n+1} \}, \] is a smooth manifold. It is homeomorphic to \( \mathbb{S}^{4n-1} \), and thus compact.