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

Abstract

The conformal Gauss maps of time-like minimal surfaces in $E^3_1$ 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 $(M, h)$, if $J$ is an $h$-reversing almost paracomplex structure of $M$ such that $\nabla J$ 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 $\nabla J$ is valued in a light-like $2n$-dimensional distribution $\mathcal{D}$ such that $(M , h, \mathcal{D} )$ is a Walker manifold and that the square norm $\parallel\!\nabla J\!\parallel^2$ of $\nabla J$ 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 $E^4_2$ such that each pair gives sections of the two time-like twistor spaces with fully light-like covariant derivatives.