We study the totally null surfaces of the neutral K¨ahler metric on certain 4-manifolds. The tangent spaces of totally null surfaces are either self-dual (α-planes) or anti-self-dual (β-planes) and so we consider α-surfaces and β-surfaces. The metric of the examples we study, which include the spaces of oriented geodesics of 3-manifolds of constant curvature, are anti-self-dual, and so it is well-known that the α-planes are integrable and α-surfaces exist. These are holomorphic Lagrangian surfaces, which for the geodesic spaces correspond to totally umbilic foliations of the underlying 3-manifold. The β-surfaces are less known and our interest is mainly in their description. In particular, we classify the β-surfaces of the neutral K¨ahler metric on T N, the tangent bundle to a Riemannian 2-manifold N. These include the spaces of oriented geodesics in Euclidean and Lorentz 3-space, for which we show that the β-surfaces are affine tangent bundles to curves of constant geodesic curvature on S 2 and H2 , respectively. In addition, we construct the β-surfaces of the space of oriented geodesics of hyperbolic 3-space.
Georgiou, Nikos & Guilfoyle, Brendan & Klingenberg, Wilhelm. (2016). Totally Null Surfaces in Neutral Kaehler 4-Manifolds. Balkan Journal of Geometry and Its Applications. 21. 27-.