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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    $$\lambda _1\ge \lambda _2&gt;0.$$  λ 1 ≥ λ 2 &gt; 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.

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