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Homogeneous geodesics of non-reductive homogeneous pseudo-Riemannian 4-manifolds

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Abstract

After obtaining an explicit description in coordinates of invariant metrics on four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds, for each of these spaces we completely describe Killing and geodesic vector fields and homogeneous geodesics through a point. In particular, this allows us to determine explicit examples of four-dimensional non-reductive pseudo-Riemannian g.o. spaces. Finally, we determine the possible holonomy Lie algebras of the invariant metrics.

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References

  1. W. Ambrose and I.M. Singer. A Theorem on Holonomy. Trans. Amer. Math. Soc., 75 (1953), 428–443.

    Article  MATH  MathSciNet  Google Scholar 

  2. L. Bérard-Bérgery and A. Ikemakhen. Sur l’holonomie des variétés pseudo-riemanniennes de signature (n, n). Bull. Soc. Math. Franc., 125 (1997), 93–114.

    Google Scholar 

  3. M. Berger. Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes. Bull. Soc. Math. Franc., 83 (1955), 279–330.

    MATH  Google Scholar 

  4. G. Calvaruso. Homogeneous structures on three-dimensional Lorentzian manifolds. J. Geom. Phys., 57 (2007), 1279–1291. Addendum: J. Geom. Phys., 58 (2008), 291–292.

    Article  MATH  MathSciNet  Google Scholar 

  5. G. Calvaruso and Z. Dusek. A n.g.o. space whose geodesics need a reparametrization. Geometry, integrability and quantization, Softex, Sofia, 2008, 167–174.

    Google Scholar 

  6. G. Calvaruso and R. Marinosci. Homogeneous geodesics of three-dimensional unimodular Lorentzian Lie groups. Mediterranean J. Math., 3 (2006), 467–481.

    Article  MATH  MathSciNet  Google Scholar 

  7. G. Calvaruso and R. Marinosci. Homogeneous geodesics of non-unimodular Lorentzian Lie groups and naturally reductive Lorentzian spaces in dimension three. Adv. Geom., 8 (2008), 473–489.

    Article  MATH  MathSciNet  Google Scholar 

  8. G. Calvaruso and A. Fino. Ricci solitons and geometry of four-dimensional nonreductive homogeneous spaces. Canad. J. Math., 64 (2012), 778–804.

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Calvaruso and A. Zaeim. Geometric structures over non-reductive homogeneous 4-spaces. Adv. Geom., 14 (2014), 191–214.

    Article  MATH  MathSciNet  Google Scholar 

  10. Z. Chen and J.A. Wolf. Pseudo-Riemannian weakly symmetric manifolds. Ann. Glob. Anal. Geom., 41 (2012), 381–390.

    Article  MATH  MathSciNet  Google Scholar 

  11. Z. Dušek. Survey on homogeneous geodesics. Note Mat., 28 (2009) [(2008) on verso], suppl. n.1, 147–168.

    MathSciNet  Google Scholar 

  12. Z. Dušek. On the reparametrization of affine homogeneous geodesics. Differential Geometry, J.A. Alvarez-Lopez and E. Garcia-Rio (Eds.),World Sci., (2009), 217–226.

    Google Scholar 

  13. Z. Dušek. The existence of homogeneous geodesics in homogeneous pseudo- Riemannian and affine manifolds. J. Geom. Phys., 60 (2010), 687–689.

    Article  MATH  MathSciNet  Google Scholar 

  14. Z. Dušek. The existence of light-like homogeneous geodesics in homogeneous Lorentzian manifolds, ArXiv: 1203.6757v1.

  15. Z. Dušek and O. Kowalski. Light-like homogeneous geodesics and the Geodesic Lemma for any signature. Publ. Math. Debrece., 71 (2007), 245–252.

    MATH  Google Scholar 

  16. Z. Dušek, O. Kowalski and Z. Vlášek. Homogeneous geodesics in homogeneous affine manifolds. Res. Math., 54 (2009), 273–288.

    Article  MATH  Google Scholar 

  17. Z. Dušek, O. Kowalski and Z. Vlášek. Homogeneous Geodesics in 3-dimensional Homogeneous Affine Manifolds. Acta Univ. Palackianae Olomucensi., 50 (2011), 29–42.

    Google Scholar 

  18. M.E. Fels and A.G. Renner. Non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Canad. J. Math., 58 (2006), 282–311.

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Galaev and T. Leistner. Recent developments in pseudo-Riemannian holonomy theory, in Handbook of Pseudo-Riemannian Geometry. IRMA Lectures in Mathematics and Theoretical Physics, (2010), 581–629.

    Google Scholar 

  20. O. Kowalski and J. Szenthe. On the existence of homogeneous geodesics in homogeneous Riemannian manifolds. Geom. Dedicat., 81 (2000), 209–214. Erratum: Geom. Dedicata, 84 (2001), 331–332.

    Article  MATH  MathSciNet  Google Scholar 

  21. O. Kowalski and L. Vanhecke. Riemannian manifolds with homogeneous geodesics. Boll. Un. Mat. Ital., 5 (1991), 189–246.

    MATH  MathSciNet  Google Scholar 

  22. O. Kowalski and Z. Vlášek. Homogeneous Riemannian manifolds with only one homogeneous geodesic. Publ. Math. Debrece., 62 (2003), 437–446.

    MATH  Google Scholar 

  23. P. Meessen. Homogeneous Lorentzian spaces whose null-geodesics are canonically homogeneous. Lett. Math. Phys., 75 (2006), 209–212.

    Article  MATH  MathSciNet  Google Scholar 

  24. P.J. Olver. Equivalence, invariants and symmetry. Cambridge University Press, 1995.

    Book  MATH  Google Scholar 

  25. J. Patera, R.T. Sharp, P. Winternitz and H. Zassenhaus. Invariants of real low dimension Lie algebras. J. Math. Phys., 17 (1976), 986–994.

    Article  MATH  MathSciNet  Google Scholar 

  26. S. Philip. Plane-wave limits and homogeneous M-theory backgrounds. Ph.D. thesis, University of Edinburgh, (2005).

    Google Scholar 

  27. J.F. Schell. Classification of 4-dimensional Riemannian spaces. J. Math. Phys., 2 (1960), 202–206.

    Article  MathSciNet  Google Scholar 

  28. R. Shaw. The subgroup structure of the homogeneous Lorentz group. Quart. J. Math. 21 (1970), 101–124.

  29. D. The. Invariant Yang-Mills connections over non-reductive pseudo-Riemannian homogeneous spaces. Trans. Amer. Math. Soc., 361 (2009), 3879–3914.

    Article  MATH  MathSciNet  Google Scholar 

  30. J.A. Wolf. Spaces of constant curvature. Sixth edition. AMS Chelsea Publishing, Providence, RI (2011).

    MATH  Google Scholar 

  31. H. Wu. On the de Rham decomposition theorem. Illinois J. Math., 8 (1964), 291–311.

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Fisica “E. De Giorgi”, Università del Salento, Prov. Lecce-Arnesano, 73100, Lecce, Italy

    Giovanni Calvaruso

  2. Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123, Torino, Italy

    Anna Fino

  3. Department of Mathematics, Payame noor University, P.O. Box 19395-3697, Tehran, Iran

    Amirhesam Zaeim

Authors
  1. Giovanni Calvaruso
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  2. Anna Fino
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  3. Amirhesam Zaeim
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Corresponding author

Correspondence to Anna Fino.

Additional information

First author partially supported by funds of the University of Salento and GNSAGA (Italy). Second author partially supported by GNSAGA (Italy).

This work was prepared during the stay of the third author at the University of Salento.

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Calvaruso, G., Fino, A. & Zaeim, A. Homogeneous geodesics of non-reductive homogeneous pseudo-Riemannian 4-manifolds. Bull Braz Math Soc, New Series 46, 23–64 (2015). https://doi.org/10.1007/s00574-015-0083-0

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  • DOI: https://doi.org/10.1007/s00574-015-0083-0

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