\({\mathcal H}\)-space with a cosmological constant. (English) Zbl 0549.53042

LeBrun, C. R.

Proc. R. Soc. Lond., Ser. A 380, 171-185 (1982).

Summary: It is demonstrated that a real-analytic 3-manifold with Riemannian conformal metric is naturally the conformal infinity of a germ-unique real-analytic 4-manifold with real-analytic Riemannian metric satisfying the self-dual Einstein equations with cosmological constant -1. Moreover, this result holds if ’Riemannian’ is replaced in the first case by ’Lorentzian’ (i.e. signature \(+--)\) and in the second case by ’pseudo- Riemannian with signature \(++--'\), or if ’real-analytic’ is replaced by ’complex-analytic’ and ’Riemannian’ is replaced by ’holomorphic’. This provides a cosmological-constant analogue of Newman’s \({\mathcal H}\)-space construction.

MSC:

53C20	Global Riemannian geometry, including pinching
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C80	Applications of global differential geometry to the sciences

Keywords:

real-analytic 3-manifold; conformal metric; conformal infinity; self-dual Einstein equations; cosmological constant; Newman’s \({\mathcal H}\)-space construction