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Durham University

Department of Mathematical Sciences


Publication details for Wilhelm Klingenberg

Georgiou, N., Guilfoyle, B. & Klingenberg, W. (2016). Totally null surfaces in neutral Kähler 4-manifolds. Balkan Journal of Geometry and Its Applications 21(1): 27-41.

Author(s) from Durham


We 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 (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 Kähler 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 S2 and H2, respectively. In addition, we construct the β-surfaces of the space of oriented geodesics of hyperbolic 3-space.