Stationary Riemannian space-times with self-dual curvature

TLDR: In this article, it was shown that for non-KSD space-times every regular compact level surface of the ψ field encloses the total NUT charge, which must be proportional to the Euler number of the surface.

Abstract: Riemannian space-times with self-dual curvature and which admit at least one Killing vector field (stationary) are examined. Such space-times can be classified according to whether a certain scalar fieldψ (which is the difference between the Newtonian and NUT potentials) reduces to a constant or not. In the former category (called here KSD) are the multi-TaubNUT and multi-instanton space-times. Nontrivial examples of the latter category have yet to be discovered. It is proved here that the static self-dual metrics are flat. It is also proved that each stationary metric for which the Newtonian and nut potentials are functionally related admits a Killing vector field relative to which the metric is KSD. It has also been proved that the regularity of theψ field everywhere implies that the metric is KSD. Finally it is proved that for non-KSD space-times every regular compact level surface of theψ field encloses the total NUT charge, which must be proportional to the Euler number of the surface.