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Null hypersurfaces in 4-manifolds endowed with a product structure

Part of: Global differential geometry

Published online by Cambridge University Press:  28 September 2023

Nikos Georgiou
Nikos Georgiou*
Affiliation:
Department of Computing and Mathematics, South East Technological University (SETU), Waterford, Ireland.
*
Email: nikos.georgiou@setu.ie
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Abstract

In a 4-manifold, the composition of a Riemannian Einstein metric with an almost paracomplex structure that is isometric and parallel defines a neutral metric that is conformally flat and scalar flat. In this paper, we study hypersurfaces that are null with respect to this neutral metric, and in particular we study their geometric properties with respect to the Einstein metric. Firstly, we show that all totally geodesic null hypersurfaces are scalar flat and their existence implies that the Einstein metric in the ambient manifold must be Ricci-flat. Then, we find a necessary condition for the existence of null hypersurface with equal nontrivial principal curvatures, and finally, we give a necessary condition on the ambient scalar curvature, for the existence of null (non-minimal) hypersurfaces that are of constant mean curvature.

Keywords

almost paracomplex structuresneutral metricsproduct of 2-manifoldsnull hypersurfaces

MSC classification

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Secondary: 53C50: Lorentz manifolds, manifolds with indefinite metrics
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 66 , Issue 1 , January 2024 , pp. 24 - 35
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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