Sections of time-like twistor spaces with light-like or zero covariant derivatives

Abstract

The conformal Gauss maps of time-like minimal surfaces in the Lorentz–Minkowski 3-space 𝐸³₁ give sections of the time-like twistor spaces associated with the pull-back bundles such that the covariant derivatives are fully light-like, that is, these are either light-like or zero, and do not vanish at any point. For an oriented neutral 4𝑛-manifold (\tilde{𝑀}, ℎ), if 𝐽 is an ℎ-reversing almost paracomplex structure of \tilde{𝑀} such that ∇𝐽 is locally given by the tensor product of a nowhere zero 1-form and an almost nilpotent structure related to 𝐽, then we will see that ∇𝐽 is valued in a light-like 2𝑛-dimensional distribution 𝒟 such that (\tilde{𝑀}, ℎ, 𝒟) is a Walker manifold and that the square norm ‖ ∇𝐽 ‖² of ∇𝐽 vanishes. We will obtain examples of ℎ-reversing almost paracomplex structures of 𝐸^4𝑛_2𝑛 as above. In addition, we will obtain all the pairs of ℎ-reversing almost paracomplex structures of 𝐸⁴₂ such that each pair gives sections of the two time-like twistor spaces with fully light-like covariant derivatives.