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

Abstract

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)), where lambda _1ge lambda _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-smooth surface in (widetilde{E(2)},g_L(lambda _1,lambda _2)) away from characteristic points and signed geodesic curvature for Euclidean 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.