Totally null surfaces in neutral Kaehler 4-manifolds
type: Item
instance of: scholarly article
zbMATH Open document ID: 1356.53027
author: Nikos Georgiou
author: Brendan Guilfoyle
author: Wilhelm Klingenberg
publication date: 2016-11-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/0810.4054
full work available at URL: B21-1ge-a87.pdf
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53B30
Mathematics Subject Classification ID: 53A25
Mathematics Subject Classification ID: 53B35
zbMATH DE Number: 6655787
zbMATH Keywords: neutral Kähler surface
zbMATH Keywords: self-duality
zbMATH Keywords: \(\alpha\)-planes
zbMATH Keywords: \(\beta\)-planes
MaRDI profile type: Publication
arXiv ID: 0810.4054
arXiv classification: math.DG
Recommended article: Null-homologous exotic surfaces in 4-manifolds
Recommended article: Surfaces with zero mean curvature vector in neutral 4-manifolds
Recommended article: Area-stationary surfaces in neutral Kähler 4-manifolds
Recommended article: Nilpotent structures of neutral 4-manifolds and light-like surfaces
Recommended article: Q3147561
Recommended article: Normalized Null hypersurfaces of Indefinite Kähler Manifolds
Recommended article: NULL SURFACES AND CONTACT GEOMETRY
Recommended article: The causal topology of neutral 4-manifolds with null boundary
Recommended article: Almost Hermitian, almost Kähler structures on neutral 4-manifolds
Recommended article: Q5135392
title: Totally null surfaces in neutral Kaehler 4-manifolds
published in: Balkan Journal of Geometry and its Applications (BJGA)
review text: The authors 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 (\(\alpha\)-planes) or anti-self-dual (\(\beta\)-planes) and so they consider \(\alpha\)-surfaces and \(\beta\)-surfaces. The metric of the examples they 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 \(\alpha\)-planes are integrable and \(\alpha\)-surfaces exist. These are holomorphic Lagrangian surfaces, which for the geodesic spaces correspond to totally umbilic foliations of the underlying 3-manifold. The \(\beta\)-surfaces are less known. The authors classify the \(\beta\)-surfaces of the neutral Kähler metric on the tangent bundle \(TN\) to a Riemannian 2-manifold \(N\). These include the spaces of oriented geodesics in Euclidean and Lorentz 3-space, for which they show that the \(\beta\)-surfaces are affine tangent bundles to curves of constant geodesic curvature on \(S^2\) and \(H^2\), respectively. In addition, they construct the \(\beta\)-surfaces of the space of oriented geodesics of hyperbolic 3-space.
reviewed by: Andreas Arvanitoyeorgos
description: scientific article; zbMATH DE number 6655787