Flat almost complex surfaces in the homogeneous nearly Kähler \(S^3\times S^3\)
type: Item
instance of: scholarly article
title: Flat almost complex surfaces in the homogeneous nearly Kähler \(S^3\times S^3\)
zbMATH Open document ID: 1393.53049
author: B. Dioos
author: Haizhong Li
author: Hui Ma
author: Luc Vrancken
published in: Results in Mathematics
publication date: 2018-04-13T00:00:00Z
review text: A nearly Kähler manifold \((\widetilde M,g,J)\) is an almost Hermitian manifold, with almost complex structure \(J\) and Levi-Civita connection \(\widetilde\nabla\) of the metric \(g\), for which the tensor field \(G=\widetilde\nabla J\) is skew-symmetric. A nearly Kähler manifold is called strictly nearly Kähler if furthermore \(\widetilde\nabla_X J=0\) for any non-vanishing tangent vector \(X\). The most well-known and simple example of a nearly Kähler manifold is the 6-dimensional sphere, whose almost complex structure \(J\) can be defined via octonionic multiplication. The only homogeneous 6-dimensional nearly Kähler manifolds are \(S^6\), \(S^3\times S^3\), \(\mathbb CP^3\), and the flag manifold \(\mathrm{SU}(3)/\mathrm{U}(1)\times \mathrm{U}(1)\). These spaces are compact 3-symmetric spaces. A natural class of submanifolds of nearly Kähler manifolds are the almost complex submanifolds, for which \(J\) sends tangent vectors to tangent ones. If \(M\) is an almost complex surface in \(S^3\times S^3\) with metric \(g\) and complex coordinate \(z=u+iv\) on \(M\), then \(\Lambda dz^2=g(P\partial_z,\partial_z)dz^2\) defines a holomorphic quadratic differential on the surface, where \(P\) is an almost product structure on \(M\). In this paper, by using a special adapted frame, the authors prove a Bonnet-type existence and uniqueness theorem for almost complex surfaces in the homogeneous nearly Kähler manifold \(S^3\times S^3\). It is shown that if \(U\subset\mathbb R^2\) is a simply connected open set endowed with a metric \(g=e^\omega dzd\bar z\), where \(z\) is a complex coordinate on \(U\), \(\omega\) is a smooth nonzero function defined on \(U\), and there exists a function \(\mu:U\to\mathbb C\) satisfying \(\omega_{z\bar z}\sinh\omega-\frac{e^{-\omega}}2|\omega_z|^2+\frac43\sinh^2\omega(1-|\mu|^2)=0\) and \(\mu_{\bar z}+\frac{\omega_{\bar z}e^\omega\mu-\omega_z\bar\mu}{2\sinh\omega}=0\), then there is an almost complex immersion \(\varphi:U\to S^3\times S^3\) wi…
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53B25
zbMATH DE Number: 6859351
zbMATH Keywords: almost complex surface
zbMATH Keywords: flat surface
zbMATH Keywords: nearly Kähler manifold
MaRDI profile type: Publication
full work available at URL: s00025-018-0784-y
OpenAlex ID: W2793201344
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DOI: 10.1007/S00025-018-0784-Y
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description: scientific article; zbMATH DE number 6859351