Totally Null Surfaces in Neutral Kaehler 4-Manifolds

Abstract

We study the totally null surfaces of the neutral Kaehler 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 Kaehler metric on $TN$, 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 $H^2$, respectively. In addition, we construct the $β$-surfaces of the space of oriented geodesics of hyperbolic 3-space.