Gauss-Bonnet Theorem in the Universal Covering Group of Euclidean Motion Group E(2) with the General Left-Invariant Metric

The universal covering group of Euclidean motion group E(2) with the general left-invariant metric is denoted by $$(\widetilde{E(2)},g_L(\lambda _1,\lambda _2)),$$( E ( 2 ) ~ , g L ( λ 1 , λ 2 ) ) , where $$\lambda _1\ge \lambda _2>0.$$λ 1 ≥ λ 2 > 0 . It is one of three-dimensional unimodular Lie groups which are classified by Milnor. In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean $$C^2$$C 2 -smooth surface in $$(\widetilde{E(2)},g_L(\lambda _1,\lambda _2))$$( E ( 2 ) ~ , g L ( λ 1 , λ 2 ) ) away from characteristic points and signed geodesic curvature for Euclidean $$C^2$$C 2 -smooth curves on surfaces. Based on these results, we get a Gauss-Bonnet theorem in the universal covering group of Euclidean motion group E(2) with the general left-invariant metric.