(Semi-)Riemannian geometry of (para-)octonionic projective planes

We use reduced homogeneous coordinates to construct and study the (semi-)Riemannian geometry of the octonionic (or Cayley) projective plane OP2OP2, the octonionic projective plane of indefinite signature OP(1,1)OP(1,1), the para-octonionic (or split octonionic) projective plane O′P2O′P2 and the hyperbolic dual of the octonionic projective plane OH2OH2. We also show that our manifolds are isometric to the (para-)octonionic projective planes defined classically by quotients of Lie groups.