CLASSIFICATION OF HOMOGENEOUS EINSTEIN METRICS ON PSEUDO-HYPERBOLIC SPACES

We classify the effective and transitive actions of a Lie group G on an n-dimensional non-degenerate hyperboloid (also called real pseudo-hyperbolic space), under the assumption that G is a closed, connected Lie subgroup of SO(n–r; r+1), the connected component of the indefinite special orthogonal group. Assuming additionally that G acts completely reducibly on ℝ , we also obtain that any G-homogeneous Einstein pseudo-Riemannian metric on a real, complex or quaternionic pseudo-hyperbolic space, or on a para-complex or para-quaternionic projective space is homothetic to either the canonical metric or the Einstein metric of the canonical variation of a Hopf pseudo-Riemannian submersion.