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

Srdjan Vukmirović; Tijana Šukilović

doi:10.37863/umzh.v72i5.645

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

DOI: https://doi.org/10.37863/umzh.v72i5.645

Анотація

Наведено повну класифікацію лівоінваріантних метрик довільної сигнатури на групі Лі, що відповідає дійсному гіперболічному просторові. Показано, що всі такі метрики мають сталу кривизну перерізу і геодезично повні лише в рімановому випадку.

Посилання

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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: https://doi.org/10.1016/j.geomphys.2015.01.005

Wolf, Joseph. Homogeneous manifolds of constant curvature. Comment. Math. Helv. 36 (1961), 112–147. DOI: https://doi.org/10.1007/BF02566896

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