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arXiv · 1508.07872

On the existence of rigid spheres in four-dimensional spacetime manifolds

Hans-Peter Gittel·Jacek Jezierski·Jerzy Kijowski
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Abstract

This paper deals with the generalization of usual round spheres in the flat Minkowski spacetime to the case of a generic four-dimensional spacetime manifold $M$. We consider geometric properties of sphere-like submanifolds in $M$ and introduce conditions on external curvature and torsion, which lead to a definition of a {\em rigid sphere}. The main result is a local existence theorem concernig such spheres. For this purpose we apply the surjective implicit function theorem. The proof is based on a detailed analysis of the linearized problem and leads to an eight-parameter family of solutions in case when the metric tensor $g$ of $M$ is from a certain neighbourhood of the flat Minkowski metric. This contribution continues the study of rigid spheres in (Class. Quantum Grav. \textbf{30} (2013), 175010, doi:10.1088/0264-9381/30/17/175010, 18 pp.).

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Hans-Peter Gittel, Jacek Jezierski, Jerzy Kijowski. 2016-10-25. On the existence of rigid spheres in four-dimensional spacetime manifolds. https://doi.org/10.1007/s10013-016-0185-z

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