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Geodesic Completeness of the Left-Invariant Metrics on ℝHn

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Ukrainian Mathematical Journal Aims and scope

We give the full classification of left-invariant metrics of an arbitrary signature on the Lie group corresponding to the real hyperbolic space. It is shown that all metrics have constant sectional curvature and that they are geodesically complete only in the Riemannian case.

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References

  1. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer Science & Business Media (2013).

  2. N. Bokan, T. Šukilović, and S. Vukmirović, “Lorentz geometry of 4-dimensional nilpotent Lie groups,” Geom. Dedicata, 177, No. 1, 83–102 (2015).

    Article  MathSciNet  Google Scholar 

  3. G. Calvaruso and A. Zaeim, “Four-dimensional Lorentzian Lie groups,” Different. Geom. Appl., 31, No. 4, 496–509 (2013).

    Article  MathSciNet  Google Scholar 

  4. M. Guediri, “Sur la complétude des pseudo-métriques invariantes àgauche sur les groupes de Lie nilpotents,” Rend. Sem. Mat. Univ. Politec. Torino, 52, 371–376 (1994).

    MathSciNet  MATH  Google Scholar 

  5. G. R. Jensen, “Homogeneous Einstein spaces of dimension four,” J. Different. Geom., 3, No. 3-4, 309–349 (1969).

    Article  MathSciNet  Google Scholar 

  6. A. Kubo, K. Onda, Y. Taketomi, and H. Tamaru, “On the moduli spaces of left-invariant pseudo-Riemannian metrics on Lie groups,” Hiroshima Math. J., 46, No. 3, 357–374 (2016).

    Article  MathSciNet  Google Scholar 

  7. J. Lauret, “Homogeneous nilmanifolds of dimension 3 and 4,” Geom. Dedicata, 68, 145–155 (1997).

    MathSciNet  MATH  Google Scholar 

  8. J. Milnor, “Curvatures of left-invariant metrics on Lie groups,” Adv. Math., 21, No. 3, 293–329 (1976).

    Article  MathSciNet  Google Scholar 

  9. K. Nomizu, “Left-invariant Lorentz metrics on Lie groups,” Osaka J. Math., 16, No. 1, 143–150 (1979).

    MathSciNet  MATH  Google Scholar 

  10. K. Nomizu, “The Lorentz–Poincaré metric on the upper half-space and its extension,” Hokkaido Math. J., 11, 253–261 (1982).

    Article  MathSciNet  Google Scholar 

  11. S. Vukmirović, “Classification of left-invariant metrics on the Heisenberg group,” J. Geom. Phys., 94, 72–80 (2015).

    Article  MathSciNet  Google Scholar 

  12. J. A. Wolf, “Homogeneous manifolds of constant curvature,” Comment. Math. Helv., 36, No. 1, 112–147 (1962).

    MathSciNet  MATH  Google Scholar 

  13. J. A. Wolf, “Isotropic manifolds of indefinite metric,” Comment. Math. Helv., 39, No. 1, 21–64 (1964).

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Belgrade, Belgrade, Serbia

    S. Vukmirović & T. Šukilović

Authors
  1. S. Vukmirović
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  2. T. Šukilović
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Correspondence to S. Vukmirović.

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 5, pp. 611–619, May, 2020.

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Vukmirović, S., Šukilović, T. Geodesic Completeness of the Left-Invariant Metrics on ℝHn. Ukr Math J 72, 702–711 (2020). https://doi.org/10.1007/s11253-020-01810-0

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  • DOI: https://doi.org/10.1007/s11253-020-01810-0

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