Totally null surfaces in neutral Kaehler 4-manifolds (Q2834824)

The authors study the totally null surfaces of the neutral Kähler metric on certain 4-manifolds. The tangent spaces of totally null surfaces are either self-dual (\(\alpha\)-planes) or anti-self-dual (\(\beta\)-planes) and so they consider \(\alpha\)-surfaces and \(\beta\)-surfaces. The metric of the examples they 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 \(\alpha\)-planes are integrable and \(\alpha\)-surfaces exist. These are holomorphic Lagrangian surfaces, which for the geodesic spaces correspond to totally umbilic foliations of the underlying 3-manifold. The \(\beta\)-surfaces are less known. The authors classify the \(\beta\)-surfaces of the neutral Kähler metric on the tangent bundle \(TN\) to a Riemannian 2-manifold \(N\). These include the spaces of oriented geodesics in Euclidean and Lorentz 3-space, for which they show that the \(\beta\)-surfaces are affine tangent bundles to curves of constant geodesic curvature on \(S^2\) and \(H^2\), respectively. In addition, they construct the \(\beta\)-surfaces of the space of oriented geodesics of hyperbolic 3-space.