Totally null surfaces in neutral Kähler 4-manifolds

Georgiou, N.; Guilfoyle, B.; Klingenberg, W.

Abstract

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.

Citation

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

• Journal Article Type: Article
• Acceptance Date: May 2, 2016
• Online Publication Date: Aug 24, 2016
• Publication Date: Jan 1, 2016
• Deposit Date: Jun 16, 2016
• Journal: Balkan journal of geometry and its applications : BJGA.
• Print ISSN: 1224-2780
• Electronic ISSN: 1843-2875
• Publisher: Balkan Society of Geometers, University Politehnica of Bucharest
• Peer Reviewed: Peer Reviewed
• Volume: 21
• Issue: 1
• Pages: 27-41
• Publisher URL: https://www.emis.de/journals/BJGA/v21n1/B21-1.htm