Einstein deformations of hyperbolic metrics

Abstract

The simplest nontrivial examples of Einstein metrics are certainly rank one symmetric spaces. We are interested in those of negative curvature, that is the hyperbolic spaces KH (m > 2), where K is the field of real numbers (R), complex numbers (C), quaternions (H) or the algebra of octonions (O); in the last case we have only the Cayley hyperbolic plane OH. They are the noncompact duals of the projective spaces KP. We normalize the metric so that the maximum of the sectional curvature is −1. We denote by d the real dimension of K (so d = 1, 2, 4 or 8) and by n = md the real dimension of KH. The boundary sphere Sn−1 of a hyperbolic space carries a rich geometric structure, namely a conformal Carnot-Caratheodory metric. Let see this first in the real and complex examples. The real hyperbolic space (with constant sectional curvature −1) is the unit ball B in R, with the metric