Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Four-dimensional Naturally Reductive Pseudo-Riemannian Homogeneous Spaces

  • De Leo, Barbara (Dipartimento di Matematica "E. De Giorgi", Universita del Salento)
  • Received : 2011.05.27
  • Accepted : 2011.08.09
  • Published : 2012.03.23
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Abstract

Our attention is turned to four-dimensional pseudo-Riemannian naturally reductive homogeneous spaces. In particular, our study leads to a complete classification of them.

Keywords

References

  1. W. Ambrose and I. M. Singer, On homogeneous Riemannian manifolds, Duke Math. J., 25(1958), 647-669. https://doi.org/10.1215/S0012-7094-58-02560-2
  2. G. Calvaruso and R. A. Marinosci, Homogeneous geodesics of non-unimodular Lorentzian Lie groups and naturally reductive Lorentzian spaces in dimension three, Adv. Geom. 8(2008), 473-489. https://doi.org/10.1515/ADVGEOM.2008.030
  3. L. A Cordero and P. E. Parker, Left-invariant Lorentzian metrics on 3-dimensional Lie groups, Rend. Mat. Appl., 17(7)(1997), 129-155.
  4. J. E. D'Atri, Geodesic spheres and symmetries in naturally reductive homogeneous spaces, Michigan Math. J., 22(1975), 71-76. https://doi.org/10.1307/mmj/1029001423
  5. J. E. D'Atri and W. Ziller, Naturally reductive metrics and Einstein metrics on compact Lie groups, Mem. Amer. Math. Soc. 18, 215, 1979.
  6. B. De Leo and R. A. Marinosci, Special Homogeneous structures on pseudo- Riemannian manifolds, JP Journal of Geometry and Topology, 8(3)(2008), 203 - 228.
  7. P. M. Gadea and J. A. Oubina, Homogeneous pseudo-Riemannian structures and homogeneous almost para-Hermitian structures, Houston J. Math., 18(3)(1992), 449- 465.
  8. P. M. Gadea and J. A. Oubina, Reductive homogeneous pseudo-Riemannian manifolds, Mh. Math., 124(1997), 17-34. https://doi.org/10.1007/BF01320735
  9. N. Haouari, W. Batat, N. Rahmani and S. Rahmani, Three-dimensional naturally reductive homogeneous Lorentzian manifolds, Mediterranean J. Math., 5(2008), 113- 131. https://doi.org/10.1007/s00009-008-0139-0
  10. S. Kobayashi and K. Nomizu, Foundations of differential geometry, II, Intersciences Publ., New York, 1969.
  11. O. Kowalski, Spaces with volume-preserving symmetries and related classes of Riemannian manifolds, Rend. Semin. Mat. Univ. Politec. Torino, Fascicolo Speciale, (1983), 131-158.
  12. O. Kowalski, Generalized symmetric spaces, Lectures Notes in Math., Springer- Verlag, Berlin, Heidelberg, New York, 1980.
  13. O. Kowalski and L. Vanhecke, Four-dimensional naturally reductive homogeneous spaces, Rend. Sem. Mat. Torino, Fasc. Speciale, (1985), 223-232.
  14. J. Milnor, Curvature of left invariant metrics on Lie groups, Adv. Math., 21(1976), 293-329. https://doi.org/10.1016/S0001-8708(76)80002-3
  15. K. Nomizu, Invariant affine connection on homogeneous spaces, Amer. J. Math., 76(1954), 33-65. https://doi.org/10.2307/2372398
  16. B. O'Neill, Semi-Riemannian Geometry, New York Academic Press, 1983.
  17. F. Tricerri and L. Vanhecke, Homogeneous structures on Riemannian manifolds, London Math. Soc. Lect. Notes 83, Cambridge Univ. Press, 1983.
  18. F. Tricerri and L. Vanhecke, Geodesic spheres and naturally reductive homogeneous spaces, Riv. Mat. Univ. Parma, 1985.
  19. H. Wu, On the de Rham decomposition theorem, Illinois J. Math., 8(1964), 291-311.
  20. H. Wu, Decomposition of Riemannian manifolds, Bull. Amer. Math. Soc., 70(1964), 610-617. https://doi.org/10.1090/S0002-9904-1964-11212-X
  21. H. Wu, Holonomy groups of indefinite metrics, Pacific J. Math., 20(1967), 351-392. https://doi.org/10.2140/pjm.1967.20.351

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  1. Four-dimensional naturally reductive pseudo-Riemannian spaces vol.41, 2015, https://doi.org/10.1016/j.difgeo.2015.04.004