A generalization of the Lorentzian splitting theorem
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 \(({\mathbb R}\times H, -dt^ 2\oplus h)\), where \((H, h)\) is a complete Riemannian manifold, provided that \((M, g)\) contains a complete time-like line.
scientific article; zbMATH DE number 6044485
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