Abstract
AbstractThe 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.
Funder
Heilongjiang Provincial Science and Technology Department
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
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