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Communications in Mathematics

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Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups

Rory Biggs
Published Online: 2018-01-11 | DOI: https://doi.org/10.1515/cm-2017-0010


We investigate the isometry groups of the left-invariant Riemannian and sub-Riemannian structures on simply connected three-dimensional Lie groups. More specifically, we determine the isometry group for each normalized structure and hence characterize for exactly which structures (and groups) the isotropy subgroup of the identity is contained in the group of automorphisms of the Lie group. It turns out (in both the Riemannian and sub-Riemannian cases) that for most structures any isometry is the composition of a left translation and a Lie group automorphism.

MSC 2010: 53C20; 53C17; 22E30

Keywords: Riemannian structures; sub-Riemannian structures; three-dimensional Lie groups


  • [1] A. Agrachev, D. Barilari: Sub-Riemannian structures on 3D Lie groups. J. Dyn. Control Syst. 18 (1) (2012) 21-44.Google Scholar

  • [2] D.V. Alekseevskiĭ: The conjugacy of polar decompositions of Lie groups. Mat. Sb. (N.S.) 84 (126) (1971) 14-26.Google Scholar

  • [3] D.V. Alekseevskiĭ: Homogeneous Riemannian spaces of negative curvature. Mat. Sb. (N.S.) 138 (1) (1975) 93-117.Google Scholar

  • [4] A. Bellaïche: The tangent space in sub-Riemannian geometry. In: A. Bellaïche, J.J. Risler (eds.), Sub-Riemannian geometry. Birkhäuser, Basel (1996) 1-78.Google Scholar

  • [5] R. Biggs, P. T. Nagy: On Sub-Riemannian and Riemannian structures on the Heisenberg groups. J. Dyn. Control Syst. 22 (3) (2016) 563-594.CrossrefWeb of ScienceGoogle Scholar

  • [6] R. Biggs, C.C. Remsing: On the classification of real four-dimensional Lie groups. J. Lie Theory 26 (4) (2016) 1001-1035.Google Scholar

  • [7] R. Biggs, C.C. Remsing: Quadratic Hamilton-Poisson systems in three dimensions: equivalence, stability, and integration. Acta Appl. Math. 148 (2017) 1-59.Web of ScienceGoogle Scholar

  • [8] R. Biggs, C.C. Remsing: Invariant control systems on Lie groups. In: G. Falcone (ed.), Lie groups, differential equations, and geometry: advances and surveys. Springer (2017) 127-181.Google Scholar

  • [9] L. Capogna, E. Le Donne: Smoothness of subRiemannian isometries. Amer. J. Math. 138 (5) (2016) 1439-1454.Google Scholar

  • [10] C. Gordon: Riemannian isometry groups containing transitive reductive subgroups. Math. Ann. 248 (2) (1980) 185-192.Google Scholar

  • [11] C.S. Gordon, E.N. Wilson: Isometry groups of Riemannian solvmanifolds. Trans. Amer. Math. Soc. 307 (1) (1988) 245-269.Google Scholar

  • [12] K.Y. Ha, J.B. Lee: Left invariant metrics and curvatures on simply connected three-dimensional Lie groups. Math. Nachr. 282 (6) (2009) 868-898.Web of ScienceGoogle Scholar

  • [13] K.Y. Ha, J.B. Lee: The isometry groups of simply connected 3-dimensional unimodular Lie groups. J. Geom. Phys. 62 (2) (2012) 189-203.Web of ScienceCrossrefGoogle Scholar

  • [14] U. Hamenstädt: Some regularity theorems for Carnot-Carathéodory metrics. J. Differential Geom. 32 (3) (1990) 819-850.Google Scholar

  • [15] V. Jurdjevic: Geometric control theory. Cambridge University Press, Cambridge (1997).Google Scholar

  • [16] I. Kishimoto: Geodesics and isometries of Carnot groups. J. Math. Kyoto Univ. 43 (3) (2003) 509-522.Google Scholar

  • [17] V. Kivioja, E. Le Donne: Isometries of nilpotent metric groups. J. Éc. Polytech. Math. 4 (2017) 473-482.Google Scholar

  • [18] A. Krasifinski, C.G. Behr, E. Schücking, F.B. Estabrook, H.D. Wahlquist, G.F.R. Ellis, R. Jantzen, W. Kundt: The Bianchi classification in the Schücking-Behr approach. Gen. Relativity Gravitation 35 (3) (2003) 475-489.Google Scholar

  • [19] E. Le Donne, A. Ottazzi: Isometries of Carnot groups and sub-Finsler homogeneous manifolds. J. Geom. Anal. 26 (1) (2016) 330-345.CrossrefGoogle Scholar

  • [20] J. Milnor: Curvatures of left invariant metrics on Lie groups. Advances in Math. 21 (3) (1976) 293-329.Google Scholar

  • [21] R. Montgomery: A tour of subriemannian geometries, their geodesics and applications. American Mathematical Society, Providence, RI (2002).Google Scholar

  • [22] G.M. Mubarakzyanov: On solvable Lie algebras. Izv. Vys¹. Uèehn. Zaved. Matematika (1963) 114-123. In RussianGoogle Scholar

  • [23] V. Patrangenaru: Classifying 3- and 4-dimensional homogeneous Riemannian manifolds by Cartan triples. Pacific J. Math. 173 (2) (1996) 511-532.Google Scholar

  • [24] P. Petersen: Riemannian geometry. Springer, New York (2006). 2nd ed.Google Scholar

  • [25] J. Shin: Isometry groups of unimodular simply connected 3-dimensional Lie groups. Geom. Dedicata 65 (3) (1997) 267-290.Web of ScienceGoogle Scholar

  • [26] L. ©nobl, P. Winternitz: Classification and identification of Lie algebras. American Mathematical Society, Providence, RI (2014).Google Scholar

  • [27] R.S. Strichartz: Sub-Riemannian geometry. J. Differential Geom. 24 (2) (1986) 221-263.Google Scholar

  • [28] A.M. Vershik, V.Y. Gershkovich: Nonholonomic dynamical systems, geometry of distributions and variational problems. In: V.I. Arnol'd, S.P. Novikov (eds.), Dynamical systems VII. Springer, Berlin (1994) pp. 1-81.Google Scholar

  • [29] E.N. Wilson: Isometry groups on homogeneous nilmanifolds. Geom. Dedicata 12 (3) (1982) 337-346.Google Scholar

About the article

Received: 2016-10-29

Accepted: 2017-08-17

Published Online: 2018-01-11

Published in Print: 2017-12-20

Citation Information: Communications in Mathematics, Volume 25, Issue 2, Pages 99–135, ISSN (Online) 2336-1298, DOI: https://doi.org/10.1515/cm-2017-0010.

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© 2018. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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