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Codimension two spacelike submanifolds of the Lorentz-Minkowski spacetime into the light cone

Part of: Local differential geometry Global differential geometry

Published online by Cambridge University Press:  17 January 2019

Luis J. Alías ,
Verónica L. Cánovas  and
Marco Rigoli
Luis J. Alías
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain ([email protected]; [email protected])
Verónica L. Cánovas
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain ([email protected]; [email protected])
Marco Rigoli
Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, I-20133, Milano, Italy ([email protected])
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Abstract

Following an original idea of Palmas, Palomo and Romero, recently developed in [12], we study codimension two spacelike submanifolds contained in the light cone of the Lorentz-Minkowski spacetime through an approach which allows us to compute their extrinsic and intrinsic geometries in terms of a single function u. As the first application of our approach, we classify the totally umbilical ones. For codimension two compact spacelike submanifolds into the light cone, we show that they are conformally diffeomorphic to the round sphere and that they are given by an explicit embedding written in terms of u. In the last part of the paper, we consider the case where the submanifold is (marginally, weakly) trapped. In particular, we derive some non-existence results for weakly trapped submanifolds into the light cone.

Keywords

spacelike submanifoldsLorentz-Minkowski spacetimelight conemean curvaturetrapped submanifolds

MSC classification

Primary: 53C40: Global submanifolds 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Secondary: 53B30: Lorentz metrics, indefinite metrics 53C50: Lorentz manifolds, manifolds with indefinite metrics
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1523 - 1553
Copyright
Copyright © Royal Society of Edinburgh 2019 

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