Asymptotically symmetric Einstein metrics. (Métriques d’Einstein asymptotiquement symétriques.) (French) Zbl 0967.53030

Biquard, Olivier

Astérisque. 265. Paris: Société Mathématique de France, 109 p. (2000).

In the work under review asymptotically symmetric Einstein metrics are studied. Asymptotically symmetric means that the curvature of these metrics at infinity is asymptotic to the curvature of a rank one symmetric space of noncompact type (that is, a hyperbolic space). Two constructions of such metrics are given. The first one relies on analysis to prove that the Einstein deformations of complex, quaternionic or octonionic symmetric spaces are in 1-1 correspondence with some Carnot-Carathéodory metrics on the boundary at infinity. In the quaternionic (resp., octonionic) case some new objects at infinity, called quaternionic (resp., octonionic) contact structures, are obtained. The second construction is twistorial: given a real analytic quaternionic structure. It is shown that a real analytic quaternionic structure is the boundary at infinity of a unique quaternionic-Kähler (and therefore Einstein), asymptotically symmetric metric, defined in a neighborhood of infinity. The geometry of quaternionic contact structures is studied.

Reviewer: Ryszard Deszcz (Wroclaw)

MSC:

53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
53-02	Research exposition (monographs, survey articles) pertaining to differential geometry
53C28	Twistor methods in differential geometry
53C26	Hyper-Kähler and quaternionic Kähler geometry, “special” geometry

Keywords:

Einstein metric; warped product; symmetric space of rank one; Carnot-Carathéodory metric; Hölder space; contact structure; quaternionic structure; octonionic structure; twistor space