Totally Null Surfaces in Neutral K¨ahler 4-Manifolds

Authors

Nikos Georgiou, Department of Mathematics Waterford Institute of Technology Waterford, Co. Waterford, Ireland.
Brendan Guilfoyle, School of Science, Technology, Engineering and Mathematics Institute of Technology, Tralee, Clash Tralee, Co. Kerry, Ireland.Follow
Wilhelm Klingenberg, Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United Kingdom.

Document Type

Article

Disciplines

Mathematics

Publication Details

© 2016 Geometry Balkan Press.

Abstract

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.

Recommended Citation

Georgiou, Nikos & Guilfoyle, Brendan & Klingenberg, Wilhelm. (2016). Totally Null Surfaces in Neutral Kaehler 4-Manifolds. Balkan Journal of Geometry and Its Applications. 21. 27-.