A generalization of the Lorentzian splitting theorem
type: Item
instance of: scholarly article
zbMATH Open document ID: 1255.53056
DOI: 10.4134/BKMS.2012.49.3.647
author: Jong-Gug Yun
published in: Bulletin of the Korean Mathematical Society
publication date: 2012-06-08T00:00:00Z
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 6044485
zbMATH Keywords: Lorentzian splitting theorem
zbMATH Keywords: time-like goedesic completeness
zbMATH Keywords: maximum principle
MaRDI profile type: Publication
full work available at URL: bkms.2012.49.3.647
OpenAlex ID: W2010392078
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title: A generalization of the Lorentzian splitting theorem
review text: The Lorentzian version of the Cheeger-Gromoll splitting theorem for Riemannian manifolds was given as a conjecture by \textit{S.-T. Yau} [Semin. differential geometry, Ann. Math. Stud. 102, 669--706 (1982; Zbl 0479.53001)]: suppose \((M, g)\) is a time-like geodesically complete Lorentzian manifold, satisfying the time-like convergence condition and admitting a unit speed time-like line, then \((M, g)\) is isometric to \(({\mathbb R}\times H, -dt^ 2\oplus h)\), where \((H, h)\) is a complete Riemannian manifold and \(({\mathbb R}, -dt^ 2)\) is represented by the given line.NEWLINENEWLINEIn a long sequence of research papers, various restricted results pertaining to this conjecture were established. Finally, \textit{R. P. A. C. Newman} [J. Differ. Geom. 31, No. 1, 163--184 (1990; Zbl 0695.53049)] succeeded in showing the validity of the original Yau conjecture.NEWLINENEWLINEIn the present paper, the author obtains a generalized Lorentzian splitting theorem by weakening the assumption of nonnegative Ricci curvature (called the time-like convergence condition in above conjecture). Specifically, he considers the two weakened conditions NEWLINE\[NEWLINE\lim\inf_{t \to \infty}\int _0^t \mathrm{Ric}(\sigma'(s), \sigma '(s))\,ds\geq 0,\tag{C1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\lim\sup _{t \to \infty}\int _0^t \max \{-\mathrm{Ric}(\sigma'(s), \sigma '(s)), 0\}\,ds\leq k ,\tag{C2}NEWLINE\]NEWLINE for a constant \(k>0\), along each time-like geodesic ray \(\sigma: [0, \infty )\to M\). The conditions (C1), (C2) are satisfied if we assume nonnegative Ricci curvature \(\mathrm{Ric}(\sigma'(s), \sigma '(s))\geq 0\) for any time-like vector \(\sigma'(s)\). By using a generalized version of the maximum principle, the author obtains the following theorem.NEWLINENEWLINELet \((M, g)\) be a time-like geodesically complete space-time of dimension \(n\geq 3\) such that the conditions (C1), (C2) are satisfied. Then \((M, g)\) splits isometrically into a product \(({\mathbb R}\times H, -dt^ …
reviewed by: Christos Baikoussis
description: scientific article; zbMATH DE number 6044485