Nilpotent structures of neutral 4-manifolds and light-like surfaces
type: Item
instance of: scholarly article
title: Nilpotent structures of neutral 4-manifolds and light-like surfaces
DOI: 10.1007/978-3-031-05379-5_2
author: Naoya Ando
publication date: 2022-12-14T00:00:00Z
review text: Let \(M\) be an oriented 4-dimensional manifold with neutral metric \(h\). Let \(U_0(\Lambda_{\pm}^2 TM)\) be bundles, contained in subbundles \(\Lambda_{\pm}^2 TM\) of rank 3 in the 2-fold power \(\Lambda^2 TM\) of \(TM\), whose fibers are light-like cones. A \((1,1)\)-tensor field \(N\) of \(M\) is an almost nilpotent structure of \(M\) if \(N\) gives a nilpotent structure of the tangent space of \(M\) at each point. The author studies almost nilpotent structures of \((M,h)\) and light-like surfaces in \((M,h)\). An almost nilpotent structure \(N\) of \(M\) gives a light-like 2-plane of the tangent space at each point of \(M\) which can be considered as a light-like two-dimensional distribution \(\mathfrak{D}\). It is shown there exists the interdependence between \(N\) and the corresponding section \(\Theta\) of either \(U_0(\Lambda_{+}^2 TM)\) or \(U_0(\Lambda_{-}^2 TM)\). Especially, the author discusses the distributions \(\mathfrak{D}\) on conditions that they are involutive. This discussion makes possibly also to study light-like surfaces in \(M\) which are integral surfaces of involutive distributions \(\mathfrak{D}\). These light-like surfaces with local horizontal lifts are characterized using the curvature tensors. It is shown that they are analogues of isotrophic minimal surfaces in Riemannian 4-manifolds. An analogy between nilpotent structures of neutral 4-manifolds from one side, and complex and para-complex structures from other side are established. For the entire collection see [Zbl 1497.53003].
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 7632937
zbMATH Keywords: neutral metric
zbMATH Keywords: nilpotent structure
zbMATH Keywords: involutive distribution
zbMATH Keywords: light-like surface
zbMATH Keywords: Walker metric
reviewed by: Neda Bokan
MaRDI profile type: Publication
full work available at URL: 978-3-031-05379-5 2
OpenAlex ID: W4313147018
cites work: COMPLEX CURVES AND ISOTROPIC MINIMAL SURFACES IN HYPERKÄHLER 4-MANIFOLDS
cites work: Surfaces in pseudo-Riemannian space forms with zero mean curvature vector
cites work: Surfaces with zero mean curvature vector in neutral 4-manifolds
cites work: Horizontality in the twistor spaces associated with vector bundles of rank 4 on tori
cites work: Hyperbolic twistor space
cites work: Conformal and minimal immersions of compact surfaces into the 4-sphere
cites work: Almost Kähler Walker 4-manifolds
cites work: Hermitian-Walker 4-manifolds
cites work: Compact complex surfaces with geometric structures related to split quaternions
cites work: Q3763283
cites work: On surfaces in four-spaces
cites work: Extremal Lorentzian surfaces with null \(r\)-planar geodesics in space forms
cites work: Q4007989
cites work: Neutral hyperkähler structures on primary Kodaira surfaces
cites work: CANONICAL FORM FOR A RIEMANNIAN SPACE WITH A PARALLEL FIELD OF NULL PLANES
description: scientific article; zbMATH DE number 7632937