{"title":"Isotropicity of surfaces in Lorentzian 4-manifolds with zero mean curvature vector","authors":"Naoya Ando","doi":"10.1007/s12188-021-00254-y","DOIUrl":null,"url":null,"abstract":"<div><p>We already have the concept of isotropicity of a minimal surface in a Riemannian 4-manifold and a space-like or time-like surface in a neutral 4-manifold with zero mean curvature vector. In this paper, based on the understanding of it, we define and study isotropicity of a space-like or time-like surface in a Lorentzian 4-manifold <i>N</i> with zero mean curvature vector. If the surface is space-like, then the isotropicity means either the surface has light-like or zero second fundamental form or it is an analogue of complex curves in Kähler surfaces. In addition, if <i>N</i> is a space form, then the isotropicity means that the surface has both the properties. If the surface is time-like and if <i>N</i> is a space form, then the isotropicity means that the surface is totally geodesic.</p></div>","PeriodicalId":50932,"journal":{"name":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2022-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s12188-021-00254-y.pdf","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s12188-021-00254-y","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 3
Abstract
We already have the concept of isotropicity of a minimal surface in a Riemannian 4-manifold and a space-like or time-like surface in a neutral 4-manifold with zero mean curvature vector. In this paper, based on the understanding of it, we define and study isotropicity of a space-like or time-like surface in a Lorentzian 4-manifold N with zero mean curvature vector. If the surface is space-like, then the isotropicity means either the surface has light-like or zero second fundamental form or it is an analogue of complex curves in Kähler surfaces. In addition, if N is a space form, then the isotropicity means that the surface has both the properties. If the surface is time-like and if N is a space form, then the isotropicity means that the surface is totally geodesic.
期刊介绍：
The first issue of the "Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg" was published in the year 1921. This international mathematical journal has since then provided a forum for significant research contributions. The journal covers all central areas of pure mathematics, such as algebra, complex analysis and geometry, differential geometry and global analysis, graph theory and discrete mathematics, Lie theory, number theory, and algebraic topology.
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