Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

On Geometric Analysis of the Dynamics of Volumetric Expansion and its Applications to General Relativity

  • 2 Accesses


In this paper, we discuss the global aspect of the geometric dynamics of volumetric expansion and its applications to the problem of the existence in the space-time of compact and complete spacelike hypersurfaces and to the global geometry of generalized Robertson–Walker space-times.

This is a preview of subscription content, log in to check access.


  1. 1.

    L. J. Alias, A. Romero, and M. Sánchez, “Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes,” Gen. Relativ. Gravit., 27, 71–84 (1995).

  2. 2.

    V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag, New York (1998).

  3. 3.

    J. K. Beem, P. E. Ehrlich, and K. L. Easley, Global Lorentzian Geometry, Marcel Dekker, New York (1996).

  4. 4.

    A. Caminha, “The geometry of closed conformal vector fields on Riemannian spaces,” Bull. Braz. Math. Soc. New Ser., 42, No. 2, 277–300 (2011).

  5. 5.

    A. Caminha, P. Souza, and F. Camargo, “Complete foliations of space forms by hypersurfaces,” Bull. Braz. Math. Soc. New Ser., 41, No. 3, 339–353 (2010).

  6. 6.

    F. Costantino and D. Thurston, “3-Manifolds efficiently bound 4-manifolds,” J. Topol., 1, No. 3, 703–745 (2008).

  7. 7.

    B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Feometry—Methods and Applications, Part 1, Grad. Texts Math., 93, Springer-Verlag, New York etc. (1992).

  8. 8.

    G. J. Galloway, “Some global aspects of compact space-times,” Arch. Math., 42, No. 2, 168–172 (1984).

  9. 9.

    L. Godinho and J. Natario, An Introduction to Riemannian Geometry with Applications to Mechanics and Relativity, Springer-Verlag, Heidelberg–New York–London (2014).

  10. 10.

    L. Guillou and A. Marin, A la Recherche de la Topologie Perdue, Birkh¨auser, Boston–Basel–Stuttgart (1986).

  11. 11.

    M. Gutierres and B. Olea, “Global decomposition of a Lorentzian manifold as a generalized Robertson–Walker space,” Differ. Geom. Appl., 27, 145–156 (2009).

  12. 12.

    S. W. Hawking and R. Penrose, “The singularities of gravitational collapse and cosmology,” Proc. Roy. Soc. Lond. A, 314, 529–548 (1970).

  13. 13.

    Sh. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, Berlin–Heidelberg–New York (1972).

  14. 14.

    Sh. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. 1, Interscience, New York–London (1963).

  15. 15.

    L. Markus, “Parallel dynamic systems,” Topology, 8, 47–57 (1969).

  16. 16.

    C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, Vol. 2, W. H. Freeman and Co., San Francisco (1973).

  17. 17.

    B. O’Neil, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, San Diego (1983).

  18. 18.

    S. Nishikawa, “On maximal spacelike hypersurfaces in Lorentzian manifold,” Nagoya Math. J., 95, 117–124 (1984).

  19. 19.

    R. Penrose, Structure of Space-Time, W. A. Benjamin, New York–Amsterdam (1968).

  20. 20.

    A. Romero, “The introduction of Bochner’s technique on Lorentzian manifolds,” Nonlin. Anal., 47, No. 5, 3047–3059 (2001).

  21. 21.

    A. Romero, R. M. Rubio, and J. J. Salamanka, “Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson–Walker spacetimes,” Class. Quantum Gravit., 30, 115007 (2013).

  22. 22.

    R. K. Sachs and H. Wu, General Relativity for Mathematicians, Springer-Verlag, New York (1977).

  23. 23.

    M. Sánchez, “On the geometry of generalized Robertson–Walker spacetimes: geodesics,” Gen. Relativ. Gravit., 30, No. 6, 915–932 (1998).

  24. 24.

    S. E. Stepanov, “Bochner’s technique for an m-dimensional compact manifold with an SL(m,R)-structure,” St. Petersburg Math. J., 10, No. 4, 703–714 (1999).

  25. 25.

    S. E. Stepanov, “An analytic method in general relativity,” Theor. Math. Phys., 122, No. 3, 402–414 (2000).

  26. 26.

    S. E. Stepanov and J. Mikeš, “The generalized Landau–Raychaudhuri equation and its applications,” Int. J. Geom. Meth. Mod. Phys., 12, No. 8, 1560026 (2015).

  27. 27.

    L. W. Tu, An Introduction to Manifolds, Springer, New York (2008).

  28. 28.

    B. Unal, “Divergence theorems in semi-Riemannian geometry,” Acta Appl. Math., 40, 173–178 (1995).

Download references

Author information

Correspondence to S. E. Stepanov.

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 146, Geometry, 2018.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Stepanov, S.E., Denezhkina, I.E. & Ovchinnikov, A.V. On Geometric Analysis of the Dynamics of Volumetric Expansion and its Applications to General Relativity. J Math Sci 245, 659–668 (2020). https://doi.org/10.1007/s10958-020-04715-2

Download citation

Keywords and phrases

  • volumetric expansion
  • vector field
  • flow
  • space-time
  • spacelike hypersurface

AMS Subject Classification

  • 53C25
  • 53Z05