We consider some generalization of a problem proposed byV. A. Toponogov for functions with nontimelike gradient on a globally hyperbolic space-time and a specific application of the positive solution of this problem to the cases of a Minkowski space-time and a de Sitter space-time of the first kind. Examples of smooth functions with timelike gradient on Lorentz manifolds are given. The authors obtain some sufficient conditions for level surfaces of functions with timelike gradient on a Lorentz manifold which guarantee that the manifold is globally hyperbolic. A description of the past and the future event horizons for timelike geodesics in a de Sitter space-time of the first kind is given. Some unsolved problems are formulated.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price includes VAT (USA)
Tax calculation will be finalised during checkout.
J. K. Beem and P. E. Erlich, Global Lorentzian Geometry (Marcel Dekker, New York–Basel, 1981; Mir, Moscow, 1985).
V. N. Berestovskiĭ, “On a Problem by V. A. Toponogov,” Mat. Tr. 13, 15 (2010) [Sib. Adv. Math. 21, 170 (2011)].
V. N. Berestovskiĭ, “On a problem by V. A. Toponogov and its generalizations,” in Relativity, Gravity and Geometry, Petrov 2010 Anniversary Symposium on General Relativity and Gravitation, November 1–6, 2010, Kazan. Contributed papers, 62 (Kazan, Kazan Univ., 2010).
V. N. Berestovskiĭ, “On a Problem by V. A. Toponogov and its generalizations,” Proc. of LobachevskiĭMath. Center, 42: LobachevskiĭReadings (Kazan, October 1–6, 2010), 58 (Kazan Mathematical Society, Kazan, 2010).
V. N. Berestovskiĭ, “On a Problem by V. A. Toponogov,” Proc. of International Scientific Conference “Operator Theory, Complex Analysis, and Mathematical Modeling,” Abstracts (Volgodonsk, Russia, July 4–8, 2011), 29 (Southern Mathematical Institute, Vladikavkaz, 2011).
A. L. Besse, Manifolds All of Whose Geodesics Are Closed (Springer-Verlag, Berlin–Heidelberg–New York, 1978;Mir, Moscow, 1981).
A. A. Fridman, Selected Works (Nauka, Moscow, 1966) [in Russian].
R. P. Geroch, “Domain of dependence,” J. Mathematical Phys. 11, 437 (1970).
D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Grossen. Lecture Notes in Mathematics, No. 55. (Springer-Verlag, Berlin–New York, 1968;Mir, Moscow, 1971).
S. W. Hawking, “The existence of cosmic time functions,” Proc. Roy. Soc. London Ser. A. N308, 433 (1968).
S. W. Hawking, “The Universe in a Nutshell” (Bantam Spectra, New York, 2001; Amfora, Saint Petersburg, 2013).
S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, London–New York, 1973; Mir, Moscow, 1977).
J. Leray, Hyperbolic Differential Equations (The Institute for Advanced Study, Princeton, N. J., 1953; Nauka Moscow, 1984).
N. V. Mitskevich, A. P. Efremov, and A. I. Nesterov, Dynamics of Fields in General Relativity (Energoatomizdat, Moscow, 1985) [in Russian].
R. Penrose, “Structure of Space-Time,” in Battelle Rencontres. 1967. Lectures in Mathematics and Physics, Chapter VII, 121 (Benjamin, New York, 1968; Mir, Moscow, 1972).
B. A. Rozenfel’d, “Non-Euclidean Geometries,” (Gosudarstv. Izdat. Tehn. -Teor. Lit., Moscow, 1955) [in Russian].
K. Nomizu, “Left-invariant Lorentz metrics on Lie groups,” Osaka. J. Math. 16 143 (1979).
S. Weinberg, Gravitation and Cosmology (John Wiley and Sons, New York–London–Sydney–Toronto, 1972; Mir, Moscow, 1975).
H. Whitney, “Differentiable manifolds,” Ann. of Math. (2) 37, 645 (1936).
Original Russian Text © V.N. Berestovskiĭ and I.A. Zubareva, 2014, published in Matematicheskie Trudy, 2014, Vol. 17, No. 2, pp. 41–60.
About this article
Cite this article
Berestovskiĭ, V.N., Zubareva, I.A. Functions with (non)timelike gradient on a space-time. Sib. Adv. Math. 25, 243–254 (2015). https://doi.org/10.3103/S1055134415040021
- globally hyperbolic space-time
- Cauchy surface
- (non)timelike gradient
- Lorentz manifold
- level surface
- de Sitter space-time of the first kind
- world line
- event horizon