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Isotropicity of surfaces in Lorentzian 4-manifolds with zero mean curvature vector

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A Correction to this article was published on 01 October 2023

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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.

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Acknowledgements

The author is grateful to the referee for valuable comments. The author was supported by Grant-in-Aid for Scientific Research (17K05221), Japan Society for the Promotion of Science.

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  1. Faculty of Advanced Science and Technology, Kumamoto University, 2-39-1 Kurokami, Kumamoto, 860-8555, Japan

    Naoya Ando

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  1. Naoya Ando
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Correspondence to Naoya Ando.

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Communicated by Vicente Cortés.

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Ando, N. Isotropicity of surfaces in Lorentzian 4-manifolds with zero mean curvature vector. Abh. Math. Semin. Univ. Hambg. 92, 105–123 (2022). https://doi.org/10.1007/s12188-021-00254-y

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