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

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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.

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References

  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).

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    MATH  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    Google Scholar 

  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).

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Correspondence to S. E. Stepanov.

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

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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

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