Twistor Theory and the Einstein Equations

R. Penrose (in Advances in twistor theory, pp. 168‒176, and in Cosmology and gravitation; Nato advanced study institute series, pp. 287‒316 (1980). New York: Plenum Press) has argued that the goal of twistor theory with regard to the vacuum Einstein equations ought to consist of some kind of unification of twistor-theoretic descriptions of anti-self-dual (a. s. d. ) and self-dual (s. d. ) space-times. S. d. space‒times currently possess a description only in terms of dual twistor space, however, rather than twistor space. In this paper, suggestions due to Penrose for providing a purely twistor space description of s. d. space‒times are investigated. It is shown how the points of certain s. d. space‒times define mappings on twistor space and the geometry of these mappings is studied. The families of mappings for two particular s. d. space‒times are presented explicitly.