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Totally null surfaces in neutral Kähler 4-manifolds

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


N. Georgiou

B. Guilfoyle


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.


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