Abstract
We classify left invariant metrics on the 4-dimensional, simply connected, unimodular Lie groups up to automorphism. When the corresponding Lie algebra is of type (R), this is equivalent to classifying the left invariant metrics up to isometry, but in general the classification up to automorphism is finer than that up to isometry. In the abelian case, all left invariant metrics are isometric. In the nilpotent case, the space of metrics can have dimension 1 or 3. In the solvable case, the dimension can be 2, 4, or 5. There are two non-solvable 4-dimensional unimodular groups, and the space of metrics has dimension 6 in both of these cases.
Similar content being viewed by others
References
Alekseevskii, D.V.: Homogeneous Riemannian spaces of negative curvature. Mat. Sb. 96(138), 87–109 (1975)
Andrada, A., Barberis, M.L., Dotti, I.G., Ovando, G.P.: Product structures on four dimensional solvable Lie algebras. Homol. Homotopy Appl 7, 9–37 (2005)
Biggs, R., Remsing, C.C.: On the classification of real four-dimensional Lie groups. J. Lie Theory 26(4), 1001–1035 (2016)
Gordon, C., Wilson, E.N.: Isometry groups of Riemannian solvmanifolds. Trans. Am. Math. Soc. 307, 245–269 (1988)
Ha, K.Y., Lee, J.B.: Left invariant metrics and curvatures on simply connected three-dimensional Lie groups. Math. Nachr. 282(6), 868–898 (2009)
Kodama, H., Takahara, A., Tamaru, H.: The space of left-invariant metrics on a Lie group up to isometry and scaling. Manuscripta Math. 135, 229–243 (2011)
Lee, J.B., Lee, K.B., Shin, J., Yi, S.: Unimodular groups of type \({\mathbb{R}}^3 {\rtimes } {\mathbb{R}}\). J. Korean Math. Soc. 5, 1121–1137 (2007)
MacCallum, M.A.H.: On the classification of the real four-dimensional Lie algebras. In: Harvey A (ed.) On Einstein’s Path. Springer, New York, pp. 299–317 (1999)
Shin, J.: Isometry groups of unimodular simply connected 3-dimensional Lie groups. Geom. Dedicata 65, 267–290 (1997)
Wall, C.T.C.: Geometric structures on compact complex analytic surfaces. Topology 25(2), 119–153 (1986)
Wolfram Research, Inc., Mathematica, Version 9.0, Champaign, IL (2013)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Van Thuong, S. Metrics on 4-dimensional unimodular Lie groups. Ann Glob Anal Geom 51, 109–128 (2017). https://doi.org/10.1007/s10455-016-9527-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-016-9527-z