Geodesic completeness of the left-invariant metrics on ${{\mathbb{R}} H^n} $

  • Srdjan Vukmirović University of Belgrade, Faculty of Mathematics, Serbia
  • Tijana Šukilović University of Belgrade, Faculty of Mathematics, Serbia

Abstract

UDC 514

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

References

Arnol’d, V. I. Mathematical methods of classical mechanics, Springer Sci. & Business Media (2013). https://books.google.com.ua/books?id=5OQlBQAAQBAJ

Bokan, N.; Šukilović, T.; Vukmirović, S. Lorentz geometry of 4-dimensional nilpotent Lie groups. Geom. Dedicata 177 (2015), 83–102. https://doi.org/10.1007/s10711-014-9980-4

Calvaruso, Giovanni; Zaeim, Amirhesam. Four-dimensional Lorentzian Lie groups. Differential Geom. Appl. 31 (2013), no. 4, 496–509. https://doi.org/10.1016/j.difgeo.2013.04.006

Guediri, M. Sur la complétude des pseudo-métriques invariantes a gauche sur les groupes de Lie nilpotents. (French) [Completeness of left-invariant pseudometrics on nilpotent Lie groups] Rend. Sem. Mat. Univ. Politec. Torino 52 (1994), no. 4, 371–376. https://doi.org/10.1007/bfb0062502

Jensen, Gary R. Homogeneous Einstein spaces of dimension four. J. Differential Geometry 3 (1969), 309–349. https://doi.org/10.4310/jdg/1214429056

Kubo, Akira; Onda, Kensuke; Taketomi, Yuichiro; Tamaru, Hiroshi. On the moduli spaces of left-invariant pseudo-Riemannian metrics on Lie groups. Hiroshima Math. J. 46 (2016), no. 3, 357–374. https://doi.org/10.32917/hmj/1487991627

Lauret, Jorge. Homogeneous nilmanifolds of dimensions $3$ and $4$. Geom. Dedicata 68 (1997), no. 2, 145–155. https://doi.org/10.1023/A:1004936725971

Milnor, John. Curvatures of left invariant metrics on Lie groups. Advances in Math. 21 (1976), no. 3, 293–329. https://doi.org/10.1016/s0001-8708(76)80002-3

Nomizu, Katsumi. Left-invariant Lorentz metrics on Lie groups. Osaka Math. J. 16 (1979), no. 1, 143–150. https://projecteuclid.org/euclid.ojm/1200771834

Nomizu, Katsumi. The Lorentz–Poincaré metric on the upper half-space and its extension. Hokkaido Math. J. 11 (1982), no. 3, 253–261. https://doi.org/10.14492/hokmj/1381757803

Vukmirović, Srdjan. Classification of left-invariant metrics on the Heisenberg group. J. Geom. Phys. 94 (2015), 72–80. https://doi.org/10.1016/j.geomphys.2015.01.005

Wolf, Joseph. Homogeneous manifolds of constant curvature. Comment. Math. Helv. 36 (1961), 112–147.

Wolf, Joseph A. Isotropic manifolds of indefinite metric. Comment. Math. Helv. 39 (1964), 21–64. https://doi.org/10.1007/BF02566943

Published
29.03.2020
How to Cite
VukmirovićS., and ŠukilovićT. “Geodesic Completeness of the Left-Invariant Metrics on ${{\mathbb{R}} H^n} $”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 5, Mar. 2020, pp. 611–619, doi:10.37863/umzh.v72i5.645.
Section
Research articles