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 (E(2)~,gL(λ1,λ2)),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\widetilde{E(2)},g_L(\lambda _1,\lambda _2)),$$\end{document} where λ1≥λ2>0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1\ge \lambda _2>0.$$\end{document} 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 C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document}-smooth surface in (E(2)~,gL(λ1,λ2))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\widetilde{E(2)},g_L(\lambda _1,\lambda _2))$$\end{document} away from characteristic points and signed geodesic curvature for Euclidean C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document}-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.


Introduction
The Gauss-Bonnet theorem is at the heart of our understanding of the relationship of the intrinsic differential geometry of a surface with its topology, which is an important component in differential geometry. In recent years, there has been an increasing interest in applications of the Gauss-Bonnet theorem in physics such as the global geometry of caustics for multiple lens planes in the impulse approximation [1], weak field limits [2], Schwarzschild black hole and Schwarzschild-like black hole in bumblebee gravity [3], the deflection angle by a NAT black hole in the weak limit approximation [4] and the four dimensional charged Einstein-Gauss-Bonnet black hole [5]. As an important geometric theorem, one of the greatest challenges scheme which is quiet useful for the proof of Gauss-Bonnet theorems in three dimensional Lie groups. In particular, we consider a sequence of Riemannian manifolds denoted by Ẽ (2), g L ( 1 , 2 ) , where g L = 1 ⊗ 1 + 2 ⊗ 2 + L ⊗ , for L > 0 .
At the heart of this approach is the fact that the intrinsic horizontal geometry does not change with L. Using the the Koszul formula, we calculate the expressions of Levi-Civita connection and Riemannian curvature in the Riemannian approximants of (Ẽ(2), g L ( 1 , 2 )) in terms of the basis {X 1 , X 2 , X 3 }. These expressions will be used to define the notions of the intrinsic curvature for curves, the intrinsic geodesic curvature of curves on surfaces and the intrinsic Gaussian curvature of surfaces away from characteristic pints. Furthermore, we derive the expressions of those curvatures in order to prove a generalized Gauss-Bonnet theorem in (Ẽ(2), g L ( 1 , 2 )).
The paper is organized in the following way. Basic notions on (Ẽ(2), g L ( 1 , 2 )) and the notions which we will use throughout the paper, such as Levi-Civita connection and Riemannian curvature in the Riemannian approximants of (Ẽ(2), g L ( 1 , 2 )) , are given in Sect. 2. In Sect. 3, we give the definitions of curvature and intrinsic curvature for Euclidean C 2 -smooth regular curves in (Ẽ(2), g L ( 1 , 2 )) and derive their expressions. In Sect. 4, we define the notions of geodesic curvature, intrinsic geodesic curvature, signed geodesic curvature and intrinsic signed geodesic curvature for Euclidean C 2 -smooth regular curves on surfaces in (Ẽ(2), g L ( 1 , 2 )) and calculate their expressions. In Sect. 5, we compute the Riemannian Gaussian curvature and the intrinsic Gaussian curvature of surfaces away from characteristic pints in (Ẽ(2), g L ( 1 , 2 )). In Sect. 6, we prove the Gauss-Bonnet theorem in (Ẽ(2), g L ( 1 , 2 )) . Finally, we summarize the main results and discuss the further work in future in Sect. 7.
2 Riemannian Approximants of (Ẽ(2), g( 1 , 2 )) In this section, some basic notions on Euclidean motion group E(2) will be introduced. We consider universal covering group of the Euclidean motion group E (2) with the general left-invariant metric (Ẽ(2), g( 1 , 2 )). The Euclidean motion group is given explictly by the following matrix group: Let Ẽ (2) denote the universal covering group of E (2). Then Ẽ (2) is isomorphic to ℝ 2 × S 1 with multiplication given by for all (x, y, ), x � , y � , � ∈ ℝ 2 × S 1 . Now, we take positive constants 1 , 2 and 3 and a left-invariant frame Then this frame satisfies the following commutation relations: Then and span X 1 , X 2 , Then H = ker . To describe the Riemannian approximants to Ẽ (2), let L > 0 and define a metric g L = 1 ⊗ 1 + 2 ⊗ 2 + L ⊗ , so that X 1 , X 2 ,X 3 ∶= L − 1 2 X 3 are orthonormal basis on T Ẽ (2) with respect to g L . Note that g = g 1 be the Riemannian metric on Ẽ (2) . The approach in this paper is to define sub-Riemannian objects as limits of horizontal objects in Ẽ (2), g L ( 1 , 2 ) , where a family of metrics g L ( 1 , 2 ) is essentially obtained as an anisotropic blow-up of the Riemannian metric g 1 ( 1 , 2 ). At the heart of this approach is the fact that the intrinsic horizontal geometry does not change with L [20]. A straightforward calculation shows the following proposition on Levi-Civita connection ∇ L on (Ẽ(2), g L ( 1 , 2 )).

3
We get the following proposition. In this section, we will compute the sub-Riemannian limit of curvature of curves in (Ẽ(2), g L ( 1 , 2 )). .
In particular, if (t) is a horizontal point of , (3.5) ) be a Euclidean C 2 -smooth regular curve in the Riemannian manifold (Ẽ(2), g L ( 1 , 2 )) , we define the intrinsic curvature ∞ of at (t) to be if the limit exists.
We introduce the following notation : for continuous functions f 1 , f 2 ∶ (0, +∞) → ℝ , Journal of Nonlinear Mathematical Physics (2022) 29:626-657 If (̇(t)) ≠ 0, by (3.1), we have If (̇(t)) = 0 and d dt ( (̇(t))) = 0, by (3.1), we get In this section, we will compute the sub-Riemannian limit of the geodesic curvature on surfaces in (Ẽ(2), g L ( 1 , 2 )) . We will say that a surface Σ ⊂ ( � E(2), g L ( 1 , 2 )) is regular if Σ is a Euclidean C 2 -smooth compact and oriented surface. In particular we will assume that there exists a Euclidean C 2 -smooth function u ∶Ẽ(2) → ℝ such that and We define the characteristic set Our computations will be local and away from characteristic points of Σ . Let us define first p ∶= X 1 u, q ∶= X 2 u, and r ∶=X 3 u. We then define In particular, p 2 +q 2 = 1 . These functions are well defined at every non-characteristic point. Let then v L is the Riemannian unit normal vector to Σ and e 1 , e 2 are the orthonormal basis of Σ . On TΣ , we define a linear transformation J L ∶ TΣ → TΣ such that For every U, V ∈ TΣ , we define ∇ Σ,L U V = ∇ L U V where ∶ TG → TΣ is the projection. Then ∇ Σ,L is the Levi-Civita connection on Σ with respect to the metric g L . By (3.6),(4.2) and we have   Moreover if (̇(t)) = 0 , then ) be a regular surface and ∶ [a, b] → Σ be a Euclidean C 2 -smooth regular curve. We define the intrinsic geodesic curvature ∞ sin X 1 +̇(t)X 2 + (̇(t))X 3 .

The Sub-Riemannian Limit of the Riemannian Gaussian Curvature of Surfaces in (Ẽ(2), g L ( 1 , 2 ))
In this section, we will compute the sub-Riemannian limit of the Riemannian Gaussian curvature of surfaces in Ẽ (2), g L ( 1 , 2 ) . We define the second fundamental form II L of the embedding of Σ into (Ẽ(2), g L ( 1 , 2 )): We have the following theorem.
Since p 2 +q 2 = 1 , we have pX ip +qX iq = 0, i = 1, 2, 3. Thus, qX 1q = −pX 1p and qX 2q = −pX 2p , and we have Next, we compute the inner product of this with v L , we obtain To compute h 12 and h 21 , using the definition of the connection, we obtain Next, we compute the inner product of this with v L . Using the product rule and the identity q Lp =p Lq , we obtain To simplify this, we find p Lqp +q Lq Since ⟨∇ e 2 v L , e 2 ⟩ L = −⟨∇ e 2 e 2 , v L ⟩ L , using the definitions of connection, identities in (2.4) and grouping terms, we have Taking the inner product with v L yields .
Under some similar simplifications to Theorem 4.3 in [20] , we get ◻ The Riemannian mean curvature H L of Σ is defined by Similar to Proposition 3.8 in [8], away from characteristic point, the horizontal mean curvature H ∞ of Σ ∈ (Ẽ(2), g L ( 1 , 2 )) is given by    Here, we used the equation as L → +∞. By In this section, we will prove the Gauss-Bonnet theorem in (Ẽ(2), g L ( 1 , 2 )) . Firstly, we consider the case of a regular curve ∶ [a, b] → (Ẽ(2), g L ( 1 , 2 )) . We define the Riemannian length measure det Journal of Nonlinear Mathematical Physics (2022) 29:626-657 When (̇(t)) = 0 , we have Proof Since similar to the proof of Lemma 6.1 in [5], we can prove When (̇(t)) ≠ 0 , we have Using the Taylor expansion, we can prove From the definition of ds L and (̇(t)) = 0 , we get Proposition 6.2 Let Σ ⊂ ( � E(2), g L ( 1 , 2 )) be a Euclidean C 2 -smooth surface, Σ = {u = 0} and d Σ,L denote the surface measure on Σ with respect to the Riemannian metric g L . Let Then Proof It is well known that We define e 1 * ∶= g L (e 1 , ⋅), e 2 * ∶= g L (e 2 , ⋅), then Therefore Recalling and the Taylor expansion we get (6.1). By (2.3), we have Similar to the proof of Theorem 4.3 in [6], we get a Gauss-Bonnet theorem in (Ẽ(2), g L ( 1 , 2 )) as following: ) be a regular surface with finitely many boundary components ( Σ) i , i ∈ {1, ⋯ , n} , given by Euclidean Proof By the discussions in [6,7], one can get the number of points satisfying on i is finite. We remark that the proof of Theorem 6.3 is based on an approximation argument by using the well known Lebesgue dominated convergence theorem. According to the conditions of Lebesgue dominated convergence theorem, a set of finitely many points can be ignored as a null set. Therefore, by Proposition 4.6, we have By the classical Gauss-Bonnet theorem for a regular surface in Σ ⊂ ( � E(2), g L ( 1 , 2 )) with boudary components given by Euclidean C 2 smooth and regular curves i , we get where K Σ,L is the Gaussian curvature of Σ, ds L and d Σ,L are defined in Proposition 6.1 and Proposition 6.2 respectively, L,s i ,Σ is the signed geodesic curvature of the ith boundary component i and (Σ) is the Euler characteristic of Σ. We assume firstly that C(Σ) is the empty set. Recalling the considerations made on the Riemannian surface measure d Σ,L , we multiply the above formula by a factor 1 √ L , then By using the Lebesgue dominated convergence theorem and let L go to the infinity, then we obtain Furthermore, one can relax the condition that the characteristic set C(Σ) is the empty set and only suppose that the characteristic set C(Σ) satisfies H 1 (C(Σ)) = 0 and that ∥ ∇ H u ∥ −1 H is locally summable with respect to the Euclidean 2-dimensional Hausdorff measure near the characteristic set C(Σ) by the similar discussions of [6]. ◻

Conclusion
This paper dealed with an interesting question of Gauss-Bonnet theorem in the universal covering group of Euclidean motion group E(2) with the general left-invariant metric from the Riemannian approximation scheme. The main result of this paper is Theorem 6.3, which is Gauss-Bonnet type theorem in the universal covering group of Euclidean motion group E(2) with the general left-invariant metric. To prove Theorem 6.3, we obtained the sub-Riemannian limit of curvature of curves, sub-Riemannian limits of geodesic curvature of curves on surfaces and the Riemannian Gaussian curvature of surfaces in the universal covering group of Euclidean motion group E(2) with the general left-invariant metric. As a future work, we plan to proceed to study Gauss-Bonnet theorems in the roto-translation group with affine connection and other three-dimensional Riemannian Lie groups which were classified in [16]. In these conditions, Gauss-Bonnet theorems can be obtained through the Riemannian approximation scheme took by Balogh et al [6,7]. Gauss-Bonnet theorem connects the intrinsic differential geometry of a surface with its topology and has many applications. Therefore, it will be interesting to extend Gauss-Bonnet theorem to other different Lie groups. We believe that the results to be obtained will have some geometric applications.