Beem, John K.; Ehrlich, Paul E. Incompleteness of timelike submanifolds with nonvanishing second fundamental form. (English) Zbl 0566.53057 Gen. Relativ. Gravitation 17, 293-300 (1985). Let f :M\(\to L^ n\) be an immersion of M as a timelike hypersurface of \(L^ n\). If the inverse image of each compact set in \(L^ n\) is compact, then M is properly immersed. If N is a unit spacelike normal to M, then the second fundamental form \(S_ N: T_ pM\times T_ pM\to {\mathbb{R}}\) need not be diagonalizable. If it is diagonalizable and positive definite on \(T_ pM\), then dim \(M\geq 3\) implies Ric (v,v)\(>0\) for all nonzero nonspacelike vectors. It follows that under these assumptions M satisfies both the generic and strong energy conditions. If M is also causally disconnected, then M is nonspacelike geodesically incomplete. Cited in 4 Documents MSC: 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics 53C40 Global submanifolds Keywords:timelike hypersurface; energy conditions; causally disconnected; geodesically incomplete PDFBibTeX XMLCite \textit{J. K. Beem} and \textit{P. E. Ehrlich}, Gen. Relativ. Gravitation 17, 293--300 (1985; Zbl 0566.53057) Full Text: DOI References: [1] Beem, J. K., and Ehrlich, P. E. (1979).Math. Proc. Cambridge Philos. Soc.,85, 161. · Zbl 0405.53042 · doi:10.1017/S0305004100055584 [2] Hawking, S. W., and Penrose, R. (1970).Proc. R. Soc. Land. Ser. A,314, 529. · Zbl 0954.83012 · doi:10.1098/rspa.1970.0021 [3] Beem, J. K., and Ehrlich, P. E. (1981).Global Lorentzian Geometry (Marcel Dekker, New York). · Zbl 0462.53001 [4] Clarke, C. J. S. (1970).Proc. R. Soc. Lond. Ser. A,314, 417. · Zbl 0197.18201 · doi:10.1098/rspa.1970.0015 [5] Greene, R. E. (1970).Isometric Embeddings of Riemannian and Pseudo-Riemannian Manifolds, Memoir No. 97 (American Mathematical Society). · Zbl 0203.24004 [6] Cheng, S. Y., and Yau, S. T. (1916).Ann. Math.,104, 407. · Zbl 0352.53021 · doi:10.2307/1970963 [7] Harris, S. G. Closed and complete spacelike hypersurfaces in Minkowski space, to be published. · Zbl 0638.53050 [8] Beem, J. K., and Ehrlich, P. E. Geodesic completeness of submanifolds in Minkowski space, to be published inGeometriae Dedicata. · Zbl 0567.53043 [9] Hawking, S. W., and Ellis, G. F. R. (1973).The Large Scale Structure of Space-time (Cambridge University Press, London). · Zbl 0265.53054 [10] O’Neill, B. (1983).Semi-Riemannian Geometry (Academic Press, New York). [11] Fialkow, A. (1938).Ann. Math.,39, 762. · Zbl 0020.06601 · doi:10.2307/1968462 [12] Magid, M. A. (1982).Proc. Am. Math. Soc.,84, 237. · doi:10.1090/S0002-9939-1982-0637176-6 [13] Gromoll, D., Klingenberg, W., and Meyer, W. (1975).Riemannsche Geometrie im Grossen (Springer-Verlag, Berlin). · Zbl 0293.53001 [14] Beem, J. K., and Parker, P. E. (1984). Values of pseudoriemannian sectional curvature,Comm. Math. Helv.,59, 319. · Zbl 0573.53014 · doi:10.1007/BF02566352 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.