Almost contact structures on the set of rational curves in a 4-dimensional twistor space (Q6144572)

In complete analogy to the Penrose correspondence between 4-dimensional Riemannian manifolds and complex 3-manifolds, the author describes a possible definition of a 4-dimensional twistor space and of a 5-dimensional complex almost contact manifold parametrizing the holomorphic deformations of the twistor line. After describing the geometry of such spaces, the author proves an analogue of the well known theorem by Atiyah, Hitchin and Singer, see [\textit{M. F. Atiyah} et al., Proc. R. Soc. Lond., Ser. A 362, 425--461 (1978; Zbl 0389.53011)], showing that under natural conditions on the curvature of the underlying manifold there exists a 4-dimensional twistor space fibering over it and satisfying all the given conditions. The author then discusses several examples. In particular, it is shown that the 4-dimensional twistor space introduced by \textit{G. Ren} and \textit{W. Wang}, [J. Geom. Phys. 183, Article ID 104699, 14 p. (2023; Zbl 1506.53062)] and the one defined by \textit{M. Itoh}, [J. Math. Phys. 43, No. 7, 3783--3797 (2003; Zbl 1060.53057)] are morally identical.