Abstract
In this paper, we study vacuum static spaces. We firstly derive a Bochner-type formula for the Weyl tensor to vacuum static space. Based on a global argument, under the condition of zero radial Weyl curvature, we then obtain a pointwise identity and use it to prove that each closed vacuum static space of dimension \(n\ge 5\) with scalar curvature \(n(n-1)\) and zero radial Weyl curvature is Bach flat.
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Acknowledgements
The author is grateful to professor Xingwang Xu for carefully reading of a preliminary version of the paper. The author also thanks the referees for critical comments and valuable suggestions that helped improve the exposition of this article.
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Ye, J. Closed Vacuum Static Spaces with Zero Radial Weyl Curvature. J Geom Anal 33, 64 (2023). https://doi.org/10.1007/s12220-022-01119-3
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DOI: https://doi.org/10.1007/s12220-022-01119-3