Weil-Einstein structures, twistor spaces and manifolds of type $$S^ 1 \times S^ 3$$. (Structures de Weyl-Einstein, espaces de twisteurs et variétés de type $$S^ 1 \times S^ 3$$.)(French)Zbl 0858.53039

This is a detailed, self-contained and very clearly written treatment of the geometry of Weyl-Einstein structures on oriented conformal manifolds, focusing on dimension 4. The exposition splits into three parts. The first one collects basic definitions and important results, in particular a detailed description of the closed Weyl-Einstein structures on compact oriented conformal manifolds. At the same time, many links to the literature and related results are given. Next, the specific features of dimension four are stressed and the general results are strengthened in this case. Then the geometry induced by a Weyl structure on the associated twistor spaces is studied in detail, linking holomorphic vector fields on the twistor space and conformal vector fields on $$M$$. The general results are further specified for self-dual structures. The final part is devoted to the study of spaces locally isometric to $$S^1 \times S^3$$, from both points of view, the Riemannian geometry and the geometry of the twistor space. Several applications involve a generalization of the development in [M. Pontecorvo, Proc. Am. Math. Soc. 113, No. 1, 177-186 (1991; Zbl 0733.32022)], and the algebraic dimension of the twistor space is shown to be zero in the generic case. A large bibliography is appended.

MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

Zbl 0733.32022
Full Text: