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

Open access


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.

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

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

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

[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.

[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.

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

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

[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.

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

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

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

[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.

[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.

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

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

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

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

[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.

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

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

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

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

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

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

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

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

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

[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.

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

Journal Information

CiteScore 2017: 0.33

SCImago Journal Rank (SJR) 2017: 0.128
Source Normalized Impact per Paper (SNIP) 2017: 0.476

Mathematical Citation Quotient (MCQ) 2016: 0.28

Target Group

researchers in the fields of: algebraic structures, calculus of variations, combinatorics, control and optimization, cryptography, differential equations, differential geometry, fuzzy logic and fuzzy set theory, global analysis, mathematical physics and number theory


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 135 135 42
PDF Downloads 61 61 25