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Inhomogeneous cosmological models with homogeneous inner hypersurface geometry

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Abstract

Space-times which allow a slicing into homogeneous spatial hypersurfaces generalize the usual Bianchi models. One knows already that in these models the Bianchi type may change with time. Here we show which of the changes really appear. To this end we characterize the topological space whose points are the 3-dimensional oriented homogeneous Riemannian manifolds; locally isometric manifolds are considered as identical.

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References

  1. Krasiński, A. (1993).Physics in an inhomogeneous universe (ZGUW Warsaw).

    Google Scholar 

  2. Krasiński, A. (1994).Acta Cosmologica 20, 67; (1995).Inhomogeneous Cosmological Models (Cambridge University Press, Cambridge), in press.

    Google Scholar 

  3. Collins, C. B. (1979).Gen. Rel. Grav. 10, 925.

    Google Scholar 

  4. Krasiński, A. (1983).Gen. Rel. Grav. 15, 673.

    Google Scholar 

  5. Krasiński, A. (1981).Gen. Rel. Grav. 13, 1021.

    Google Scholar 

  6. Szekeres, P. (1975).Commun. Math. Phys. 41, 55.

    Google Scholar 

  7. Schmidt, H.-J. (1982).Astron. Nachr. 303, 283.

    Google Scholar 

  8. Schmidt, H.-J. (1982).Astron. Nachr. 303, 227.

    Google Scholar 

  9. Wolf, J. (1967).Spaces of constant curvature (McGraw-Hill, New York).

    Google Scholar 

  10. Karlhede, A., and MacCallum, M. A. H. (1982).Gen. Rel. Grav. 14, 673.

    Google Scholar 

  11. Szafron, D. (1981).J. Math. Phys. 22, 543.

    Google Scholar 

  12. Bona, C., and Coll, B. (1992).J. Math. Phys. 33, 267.

    Google Scholar 

  13. Paiva, F., Rebouças, M., and MacCallum, M. A. H. (1993).Class. Quant. Grav. 10, 1165.

    Google Scholar 

  14. Schmidt, H.-J. (1988).J. Math. Phys. 29, 1264.

    Google Scholar 

  15. Schmidt, H.-J. (1987).J. Math. Phys. 28, 1928.

    Google Scholar 

  16. Rainer, M. (1994). “The topology of the space of real Lie algebras up to dimension 4 with applications to homogeneous cosmological models.” Ph. D. thesis, Universität Potsdam.

  17. Segal, I. (1951).Duke Math. J. 18, 221.

    Google Scholar 

  18. Conatser, C. (1972).J. Math. Phys. 13, 196.

    Google Scholar 

  19. Levy-Nahas, M. (1967).J. Math. Phys. 8, 1211.

    Google Scholar 

  20. Saletan, E. (1961).J. Math. Phys. 2, 1.

    Google Scholar 

  21. Schempp, W. (1986).Harmonic analysis on the Heisenberg nilpotent Lie group (Wiley. New York).

    Google Scholar 

  22. Kaplan, A. (1983).Bull. Lond. Math. Soc. 15, 35.

    Google Scholar 

  23. Bianchi, L. (1897).Mem. della Soc. Italiana delle Scienze Ser. 3a 11, 267.

    Google Scholar 

  24. Bianchi, L. (1918).Lezioni Sulla Teoria Dei Gruppi Continui Finiti Di Trasformazioni (Spoerri, Pisa).

    Google Scholar 

  25. Lie, S., and Engel, F. (1888).Theorie der Transformationsgruppen (Teubner, Leipzig). (Part 3 of this textbook, which contains the classification of the 3-dimensional Lie algebras, appeared in 1893. The 3 volumes together have more than 2000 pages. Sometimes, this textbook is cited without the second author.)

    Google Scholar 

  26. Lie, S. (1891).Differentialgleichungen (Chelsea, Leipzig).

    Google Scholar 

  27. Lee, H. C. (1947).J. Math. Pures et Appl. 26, 251.

    Google Scholar 

  28. Ellis, G. F. R., and MacCallum, M. A. H. (1969).Commun. Math. Phys. 12, 108.

    Google Scholar 

  29. Estabrook, F., Wahlquist, H., and Behr, C. (1968).J. Math. Phys. 9, 497.

    Google Scholar 

  30. Davies, P., and Twamley, J. (1993).Class. Quant. Grav. 10, 931.

    Google Scholar 

  31. Ambrose, W., and Singer, I. (1958).Duke Math. J. 25, 647.

    Google Scholar 

  32. Tricerri, F., and Vanhecke, L. (1983).Homogeneous structures on Riemannian manifolds (Cambridge University Press, Cambridge).

    Google Scholar 

  33. Christodoulakis, T., and Korfiatis, E. (1992).J. Math. Phys. 33, 2868.

    Google Scholar 

  34. Vranceanu, G. (1964).Leçons de Géométrie différentielle (Hermann, Paris).

    Google Scholar 

  35. Milnor, J. (1976).Adv. Math. 21, 293.

    Google Scholar 

  36. Reiß, A. (1993). “Krümmungsinvarianten in 3-dimensionalen homogenen Riemannschen Räumen.” Staatsexamensarbeit, Universität Potsdam.

  37. Slansky, R. (1981).Phys. Rep. 79, 1.

    Google Scholar 

  38. Ziller, W. (1982).Math. Ann. 259, 351.

    Google Scholar 

  39. Koike, T., Tanimoto, M., and Hosoya, A. (1994).J. Math. Phys. 35, 4855.

    Google Scholar 

  40. Bona, C., and Coll, B. (1994).J. Math. Phys. 35, 873.

    Google Scholar 

  41. Schmidt, H.-J. (1994). “Why do all the curvature invariants of a gravitational wave vanish?” Preprint Universität Potsdam Math 94/03, gr-qc/9404037. To appear inNew frontiers in gravitation, G. Sardanashvili, R. Santilli, eds.

  42. Nomizu, K. (1979).Osaka J. Math. 16, 143.

    Google Scholar 

  43. Alias, L., Romero, A., and Sánchez, M. (1995).Gen. Rel. Grav. 27, 71.

    Google Scholar 

  44. Lachieze-Rey, M., and Luminet, J.-P. (1995).Phys. Rep. 254, 13.

    Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematik, Projektgruppe Kosmologie, D-14415, Universität Potsdam, PF 601553, Potsdam, Germany

    Martin Rainer & Hans-Jürgen Schmidt

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  1. Martin Rainer
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  2. Hans-Jürgen Schmidt
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Rainer, M., Schmidt, HJ. Inhomogeneous cosmological models with homogeneous inner hypersurface geometry. Gen Relat Gravit 27, 1265–1293 (1995). https://doi.org/10.1007/BF02153317

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