Left invariant metrics and curvatures on simply connected three‐dimensional Lie groups

Abstract

For each simply connected three-dimensional Lie group we determine the automorphism group, classify the left invariant Riemannian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. Thereby we obtain the principal Ricci curvatures, the scalar curvature and the sectional curvatures as functions of left invariant metrics on the three-dimensional Lie groups. Our results improve a bit of Milnor's results of [7] in the three-dimensional case, and Kowalski and Nikvcević's results [6, Theorems 3.1 and 4.1].

Group Elastic Symmetries Common to Continuum and Discrete Defective Crystals

“…where the values of β 2 , β 3 , γ 2 , γ 3 are as in (5.5), then conditions necessary and sufficient that G = D are that 12) where the components of τ i , i = 1, 2, 3, 4, are τ 1i τ 2i…”

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

“…[12]; see also [23]). Any left-invariant Riemannian structure on a simply connected three-dimensional Lie group is L-isometric, up to rescaling, to exactly one of the following Riemannian structures:…”

“…For instance, the structure (6), β = 1 on SE(1, 1) also has principle Ricci curvatures λ 1 = λ 2 = 0 and λ 3 = −1. However, for this structure the trace of S is equal to 1; thus this structure is not isometric to (12) for any β 1 > 1 (as 1 < β 2 1 +1 2β1 for β 1 > 1).…”

Control Affine Systems on Semisimple Three-Dimensional Lie Groups

“…In order to classify the affine subspaces of g, we require the (group of) automorphisms of g. These are well known (see, e.g., [7], [8], [14]); a summary is given in table 1. For each of the two Lie algebras, we construct class representatives (by considering the action of automorphisms on a typical affine subspace).…”