# The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss–Bonnet Theorem in the Rototranslation Group

Authors: Haiming Liu; Jiajing Miao; Wanzhen Li; Jianyun Guan
Journal: Journal of Mathematics
 (2021)

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## Abstract

The rototranslation groupℛT is the group comprising rotations and translations of the Euclidean plane which is a 3-dimensional Lie group. In this paper, we use the Riemannian approximation scheme to compute sub-Riemannian limits of the Gaussian curvature for a Euclidean C2-smooth surface in the rototranslation group away from characteristic points and signed geodesic curvature for Euclidean C2-smooth curves on surfaces. Based on these results, we obtain a Gauss–Bonnet theorem in the rototranslation group.


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## Body

## 1. Introduction

The Gauss–Bonnet theorem connects the intrinsic differential geometry of a surface with its topology and has many applications in physics and mathematics. For example, Petters used the Gauss–Bonnet theorem to study the global geometry of caustics for multiple lens planes in the impulse approximation [1]. Gibbons and Werner showed that it is possible to calculate the deflection angle in weak field limits using the Gauss–Bonnet theorem and the optical geometry [2]. In this method, they found that the focusing of the light rays emerges as a topological effect. In 2018, Övgün et al. used the Gauss–Bonnet theorem to obtain the deflection angle by the photons coupled to the Weyl tensor in a Schwarzschild black hole and Schwarzschild-like black hole in bumblebee gravity in the weak limit approximation [3]. Their computations about the weak gravitational lensing of the Kerr-MOG black hole utilized the method of Gauss–Bonnet first prescribed by Gibbons and Werner [2], which reveals the ignored role of topology in gravitational lensing. In 2019, they studied the weak gravitational lensing by the Kerr-MOG black hole and showed that the MOG effect could be taken into account in the gravitational lensing experiment [4]. In 2020, they applied the RVB method, which considers the topological fractions together with the Gauss–Bonnet theorem and different spacetimes including the nonasymptotically flat ones. This approach showed that Hawking radiation possesses a topological effect coming from the Euler characteristic of the spacetime. The Ricci scalar of the spacetime encodes all the information about the spacetime, which means that it can derive the temperature of the black hole with the Euler characteristic of the metric [5]. They also employed the Gauss–Bonnet theorem to compute the deflection angle by a NAT black hole in the weak limit approximation [6]. In 2021, Chen et al. investigated the photon sphere, shadow, and QNMs of the four-dimensional charged Einstein–Gauss–Bonnet black hole [7]. In this paper, we focus on the Gauss–Bonnet theorem in the rototranslation group.The rototranslation group,ℛT, is the group of Euclidean rotations and translations of the plane equipped with a particular sub-Riemannian metric. More precisely, ℛT is a three-dimensional topological manifold diffeomorphic to ℝ2×S1 with coordinates x,y,θ. The sub-Riemannian geometry of the rototranslation group is in contrast to the well-known case of the Heisenberg group in mathematics, and it provides geometrical models in mechanics and robotics [8, 9]. In [10], the rototranslation group ℛT and its universal cover G were introduced. The main theorem states that a straight ruled surface in G is horizontally minimal. Among more recent works, data representations in orientation scores as a function on the rototranslation group have been used for template matching with cross-correlation [11]. Bekkers et al. recognized a curved geometry on the position-orientation domain, which they identified with the rototranslation group. Templates were then optimized in a B-spline basis, and smoothness was defined with respect to the curved geometry. In [12], illusory patterns were identified by a suitable modulation of the geometry of the rototranslation group and computed as the associated geodesics via the fast marching algorithm. In [13, 14], Balogh et al. used a Riemannian approximation scheme to define a notion of the intrinsic Gaussian curvature for a Euclidean C2-smooth surface in the Heisenberg group H1 away from characteristic points and a notion of the intrinsic signed geodesic curvature for Euclidean C2-smooth curves on surfaces. These results were then used to prove a Heisenberg version of the Gauss–Bonnet theorem. They proposed an interesting question to understand to what extent similar phenomena hold in other sub-Riemannian geometric structures. In [15–17], Wang and Wei solved this problem for the affine group, the group of rigid motions of the Minkowski plane, the BCV spaces, the twisted Heisenberg group, and the Lorentzian Heisenberg group. Their approach is to define sub-Riemannian objects as limits of horizontal objects in G,gL, where a family of metrics gL is essentially obtained as an anisotropic blowup of the Riemannian metric g1. At the heart of this approach is the fact that the intrinsic horizontal geometry does not change with L. In this paper, we try to solve the above problem for the rototranslation group. We compute sub-Riemannian limits of the Gaussian curvature for a Euclidean C2-smooth surface in the rototranslation group away from characteristic points and signed geodesic curvature for Euclidean C2-smooth curves on surfaces. We also obtain a Gauss–Bonnet theorem in the rototranslation group.In Section2, we provide a short introduction to the rototranslation group and the notion which we will use throughout the paper, such as the Levi-Civita connection and curvature in the Riemannian approximants of the rototranslational group. In Section 3, we compute the sub-Riemannian limit of the curvature of curves in the rototranslation group. In Sections 4 and 5, we compute sub-Riemannian limits of the geodesic curvature of curves on surfaces and the Riemannian Gaussian curvature of surfaces in the rototranslation group. In Section 6, we obtain the Gauss–Bonnet theorem in the rototranslation group. In Section 7, we summarize this paper as conclusions.


## 2. Levi-Civita Connection and Curvature in the Riemannian Approximants of the Rototranslation Group

In this section, some basic notions in the rototranslation group will be introduced. The rototranslation groupℛT is the group comprising rotations and translations of the Euclidean plane. It is a 3-dimensional Lie group, isomorphic to ℝ2×S1 with multiplication given by(1)x,y,θ∗x′,y′,θ′=x+x′cosθ−y′sinθ,y+x′sinθ+y′cosθ,θ+θ′,for all x,y,θ,x′,y′,θ′∈ℝ2×S1. In this model, the natural element of ℛT is 0,0,0, and the inverse element of x,y,θ is −xcosθ−ysinθ,xsinθ−ycosθ,−θ.Let(2)X1=cosθ∂∂x+sinθ∂∂y,X2=∂∂θ,X3=sinθ∂∂x−cosθ∂∂y,with brackets(3)X1,X2=X3,X2,X3=X1,X1,X3=0.Then,(4)∂∂x=cosθX1+sinθX3,∂∂y=sinθX1−cosθX3,∂∂θ=X2,and spanX1,X2,X3=TℛT. Let H=spanX1,X2 be the horizontal distribution on ℛT. Let ω1=cosθdx+sinθdy,ω2=dθ,andω=sinθdx−cosθdy. Then, H=kerω. To describe the Riemannian approximants to ℛT, for the constant L>0, we define a metric gL=ω1⊗ω1+ω2⊗ω2+Lω⊗ω so that X1,X2,X3˜≔L−1/2X3 are orthonormal bases on TℛT with respect to gL. It is easy to check that g=g1 is the Riemannian metric on ℛT.To compute the sectional, Ricci, and scalar curvatures of the rototranslation group with respect togL, we use the Levi-Civita connection ∇L on ℛT,gL. A straightforward calculation shows the following proposition.Proposition 1.
LetℛT be the rototranslation group, relative to the coordinate frame X1,X2,X3˜; then, the Levi-Civita connection on ℛT is given by(5)∇XjLXj=0,1≤j≤3,∇X1LX2=L−12LX3,∇X2LX1=−L+12LX3,∇X1LX3=1−L2X2,∇X3LX1=−L+12X2,∇X2LX3=1+L2X1,∇X3LX2=−1+L2X1.Proof.
It follows from a direct application of the Koszul identity, which here simplifies(6)2∇XiLXj,XkL=Xi,Xj,XkL−Xj,Xk,XiL+Xk,Xi,XjL,where i,j,k=1,2,3. By (3) and (6), we have(7)2∇XjLXj,XkL=Xj,Xj,XkL−Xj,Xk,XjL+Xk,Xj,XjL=−Xj,Xk,XjL+Xk,Xj,XjL=2Xk,Xj,XjL.
Whenj=1, we compute ∇X1LX1,XkL=Xk,X1,X1L. It follows that ∇X1LX1,X1L=0, ∇X1LX1,X2L=X2,X1,X1L=−X3,X1L=0, and ∇X1LX1,X3L=X3,X1,X1L=0. Hence, ∇X1LX1=0. Similarly, ∇X2LX2=0 and ∇X3LX3=0. By the following equation,(8)2∇X1LX2,XkL=X1,X2,XkL−X2,Xk,X1L+Xk,X1,X2L=X3,XkL−X2,Xk,X1L+Xk,X1,X2L,we get ∇X1LX2=L−1/2LX3. Other cases follow the similar way.Define the curvature of the connection∇L by(9)RLX,YZ=∇XL∇YLZ−∇YL∇XLZ−∇X,YLZ.We get the following proposition.Proposition 2.
LetℛT be the rototranslation group; then,(10)RLX1,X2X1=3L2−2L−14LX2,RLX1,X2X2=−3L2+2L+14LX1,RLX1,X2X3=0,RLX1,X3X1=−L2+2L−14LX3,RLX1,X3X2=0,RLX1,X3X3=L2−2L+14X1,RLX2,X3X1=0,RLX2,X3X2=−L2−2L+34LX3,RLX2,X3X3=L2+2L−34X2.Proof.
It is a direct computation using(11)RLX,YZ=∇XL∇YLZ−∇YL∇XLZ−∇X,YLZ.
Taking(12)RLX1,X2X1=∇X1L∇X2L−∇X2L∇X1L−∇X1,X2LX1,for example, we compute(13)∇X1L∇X2LX1=−L+12L∇X1LX3=L2−14LX2,∇X2L∇X1LX1=0,∇X1,X2LX1=∇X3LX1=−L+12X2.
Hence,(14)RLX1,X2X1=L2−14LX2−−L+12X2=3L2−2L−14LX2.
We compute the sectional curvatures of the two planes spanned by the basis vectorsX˜i and X˜j: Kij=RLX˜i,X˜jX˜i,X˜jL, where X˜i=Xi for i=1,2 and X˜3=L−1/2X3.(15)K12=RLX˜1,X˜2X˜1,X˜2L=3L2−2L−14L,K13=RLX˜1,X˜3X˜1,X˜3L=−L2+2L−14L,K23=RLX˜2,X˜3X˜2,X˜3L=−L2−2L+34L.
In fact, the full Riemannian curvature tensorRijklL=RLX˜i,X˜jX˜k,X˜lL is(16)RijklL=3L2−2L−14Lifijkl=1212or2121,−3L2+2L+14Lifijkl=1221or2112,−L2+2L−14Lifijkl=1313,3131,2323or3232,L2−2L+14Lifijkl=1331,3113,2332or3223,0,otherwise.
In order to compute the Kretschmann scalar, fromRLijkl=gllgkkgjjgiiRijklL, it follows that we can write(17)RLijkl=3L2−2L−14Lifijkl=1212or2121,−3L2+2L+14Lifijkl=1221or2112,−L2+2L−14Lifijkl=1313,3131,2323or3232,L2−2L+14Lifijkl=1331,3113,2332or3223,0,otherwise.
Recall that the Kretschmann scalar is defined byK=RijklLRLijkl. We calculate that(18)K=4R1212L2+R1313L2+R2323L2=11L4−20L3+10L2−4L+34L2.
Next, the Ricci curvatureRici=Ki1+Ki2+Ki3 is(19)Ric1=L2−12L,Ric2=−L+12L−1,Ric3=−L2+12L,while the scalar curvature(20)σ=Ric1+Ric2+Ric3=−L+12L−1.
It can be observed that the Kretschmann scalar and the sectional, Ricci, and scalar curvatures all diverge asL⟶∞.


## 3. The Sub-Riemannian Limit of the Curvature of Curves in the Rototranslation Group

In this section, we will compute the sub-Riemannian limit of the curvature of curves in the rototranslation group.Definition 1.
Letγ:a,b⟶ℛT,gL be a Euclidean C1-smooth curve; we say that γ is regular if γ˙≠0 for every t∈a,b. Moreover, we say that γt is a horizontal point of γ if(21)ωγ˙t=sinθdx−cosθdyγ˙1t∂∂x+γ˙2t∂∂y+θ˙t∂∂θ=γ˙1tsinθ−γ˙2tcosθ=0,where γt=γ1t,γ2t,θt.Definition 2.
Letγ:a,b⟶ℛT,gL be a Euclidean C2-smooth regular curve in the Riemannian manifold ℛT,gL. The curvature κγL of γ at γt is defined as(22)κγL≔∇γ˙Lγ˙L2γ˙L4−∇γ˙Lγ˙,γ˙L2γ˙L6.Proposition 3.
Letγ:a,b⟶ℛT,gL be a Euclidean C2-smooth regular curve in the Riemannian manifold ℛT,gL. Then,(23)κγL=γ¨1tcosθ+γ¨2tsinθ+θ˙t−γ˙1sinθ+γ˙2cosθ+Lωγ˙t2+γ˙1tcosθ+γ˙2tsinθωγ˙t1−L+θ..t2+Lγ˙1tcosθ+γ˙2tsinθθ.t−1L+ddtωγ˙t2×γ˙1tcosθ+γ˙2tsinθ2+θ.t2+Lωγ˙t2−2−γ˙1tcosθ+γ˙2tsinθγ¨1tcosθ+γ¨2tsinθ+θ.t−γ˙1sinθ+γ˙2cosθ+Lωγ˙t+θ.tγ˙1tcosθ+γ˙2tsinθωγ˙t1−L+θ..t+Lωγ˙tγ˙1tcosθ+γ˙2tsinθθ.t−1L+ddtωγ˙t2×γ˙1tcosθ+γ˙2tsinθ2+θ.t2+Lωγ˙t2−31/2.
In particular, ifγt is a horizontal point of γ,(24)kγL=γ¨1tcosθ+γ¨2tsinθ+θ˙t−γ˙1sinθ+γ˙2cosθ2+θ..t2+Lγ˙1tcosθ+γ˙2tsinθθ.t−1L+ddtωγ˙t2⋅γ˙1tcosθ+γ˙2tsinθ2+θ.t2−2−γ˙1tcosθ+γ˙2tsinθγ¨1tcosθ+γ¨2tsinθ+θ.t−γ˙1sinθ+γ˙2cosθ+θ.tθ..t2⋅γ˙1tcosθ+γ˙2tsinθ2+θ.t2−31/2.Proof.
By (4), we have(25)γ˙t=γ˙t∂∂x+γ˙2t∂∂y+θ˙t∂∂θ=γ˙1tcosθX1+sinθX3+γ˙2tsinθX1−cosθX3+θ˙tX2=γ˙1tcosθ+γ˙2tsinθX1+θ˙tX2+γ˙1tsinθ−γ˙2tcosθX3=γ˙1tcosθ+γ˙2tsinθX1+θ˙tX2+ωγ˙1tX3.
By Proposition1 and (25), we have(26)∇γ˙LX1=γ˙1tcosθ+γ˙2tsinθ∇X1LX1+θ˙t∇X2LX1+ωγ˙t∇X3LX1=θ˙t−L+12LX3+ωγ˙t−L+12X2,∇γ˙LX2=γ˙1tcosθ+γ˙2tsinθ∇X1LX2+θ˙t∇X2LX2+ωγ˙t∇X3LX2=γ˙1tcosθ+γ˙2tsinθL−12LX3+ωγ˙t−1+L2X1,∇γ˙LX3=γ˙1tcosθ+γ˙2tsinθ∇X1LX3+θ˙t∇X2LX3+ωγ˙t∇X3LX3=γ˙1tcosθ+γ˙2tsinθ1−L2X2+θ˙t1+L2X1.
By (25) and (26), we have(27)∇γ˙Lγ˙=∇γ˙Lγ˙1tcosθ+γ˙2tsinθX2+θ˙tX2+ωγ˙tX3=γ˙γ˙1tcosθ+γ˙2tsinθX1+γ˙1tcosθ+γ˙2tsinθ∇γ˙LX1+γ˙θ˙tX2+θ˙t∇γ˙LX2+γ˙ωγ˙tX3+ωγ˙t∇γ˙LX3=γ¨1tcosθ+γ¨2tsinθ+θ˙t−γ˙1sinθ+γ˙2cosθ+Lωγ˙tX1+γ˙1tcosθ+γ˙2tsinθωγ˙t1−L+θ¨tX2+γ˙1tcosθ+γ˙2tsinθθ˙t−1L+ddtωγ˙tX3.
By (22) and (25), we get(28)∇γ˙Lγ˙L2=γ¨1tcosθ+γ¨2tsinθ+θ˙t−γ˙1tsinθ+γ˙2tcosθ+Lωγ˙t2+γ˙1tcosθ+γ˙2tsinθωγ˙t1−L+θ¨t2+Lγ˙1tcosθ+γ˙2tsinθθ˙t−1L+ddtωγ˙t2,γ˙L4=γ˙1tcosθ+γ˙2tsinθ2+θ˙t2+Lωγ˙t22,∇γ˙Lγ˙,γ˙L2=γ˙1tcosθ+γ˙2tsinθ=γ¨1tcosθ+γ¨2tsinθ+θ˙t−γ˙1sinθ+γ˙2cosθ+Lωγ˙t=+θ˙tγ˙1tcosθ+γ˙2tsinθωγ˙t1−L+θ¨t=+Lωγ˙tγ˙1tcosθ+γ˙2tsinθθ˙t−1L+ddtωγ˙t2,γ˙L6=γ˙1tcosθ+γ˙2tsinθ2+θ˙t2+Lωγ˙t23.
By the definition ofκγL, we get Proposition 3.Definition 3.
Letγ:a,b⟶ℛT,gL be a Euclidean C2-smooth regular curve in the Riemannian manifold ℛT,gL; we define the intrinsic curvature κγ∞ of γ at γt to be(29)κγ∞≔limL⟶∞κγL,if the limit exists.
We introduce the following notation: for continuous functionsf1,f2:0,+∞⟶ℝ,(30)f1L∼f2L,asL⟶+∞⇔limL⟶∞f1Lf2L=1.Proposition 4.
Letγ:a,b⟶ℛT,gL be a Euclidean C2-smooth regular curve in the Riemannian manifold ℛT,gL. Then,(31)κγ∞=θ˙2+γ˙1cosθ+γ˙2sinθ2ωγ˙tifωγ˙t≠0.(32)κγ∞=γ¨1cosθ+γ¨2sinθ+θ˙−γ˙1sinθ+γ˙2cosθ2+θ¨2⋅γ˙1cosθ+γ˙2sinθ2+θ˙2−2−γ˙1cosθ+γ˙2sinθγ¨1cosθ+γ¨2sinθ+θ˙−γ˙1sinθ+γ˙2cosθ+θ˙θ¨2⋅γ˙1cosθ+γ˙2sinθ2+θ˙2−31/2ifωγ˙t=0andddtωγ˙t=0,limL⟶∞κγLL=d/dtωγ˙tγ˙1cosθ+γ˙2sinθ2+θ˙2ifωγ˙t=0andddtωγ˙t≠0.Proof.
Whenωγ˙t≠0, we have(33)∇γ˙LγL2∼ωγ˙t2θ˙2+γ˙12cos2θ+γ˙22sin2θ+2γ˙1γ˙2cosθsinθL2∼ωγ˙t2θ˙2+γ˙1cosθ+γ˙2sinθ2L2asL⟶+∞,γ˙L2∼Lωγ˙t2,∇γ˙Lγ˙,γ˙L2∼OL2asL⟶+∞.
Therefore,(34)∇γ˙Lγ˙L2γ˙L4⟶θ˙2+γ˙1cosθ+γ˙2sinθ2ωγ˙t2asL⟶+∞,∇γ˙Lγ˙,γ˙L2γ˙L6⟶0asL⟶+∞.
Ifωγ˙t≠0, by (22), we have(35)κγ∞=θ˙2+γ˙1cosθ+γ˙2sinθ2ωγ˙t.
By (24) and d/dtωγ˙t=0, we have(36)κγ∞=γ¨1cosθ+γ¨2sinθ+θ˙−γ˙1sinθ+γ˙2cosθ2+θ¨2⋅γ˙1cosθ+γ˙2sinθ2+θ˙2−2−γ˙1cosθ+γ˙2sinθγ¨1cosθ+γ¨2sinθ+θ˙−γ˙1sinθ+γ˙2cosθ+θ˙θ¨2⋅γ˙1cosθ+γ˙2sinθ2+θ˙2−31/2.
Whenωγ˙t=0 and d/dtωγ˙t≠0, we have(37)∇γ˙LγL2∼Lddtωγ˙t2asL⟶+∞,γ˙L2=γ˙1cosθ+γ˙2sinθ2+θ˙2,∇γ˙Lγ˙,γ˙L2∼O1asL⟶+∞.
Ifωγ˙t=0 and d/dtωγ˙t=0, by (22), we get(38)limL⟶∞κγLL=d/dtωγ˙tγ˙1cosθ+γ˙2sinθ2+θ˙2.


## 4. The Sub-Riemannian Limit of the Geodesic Curvature of Curves on Surfaces in the Rototranslation Group

In this section, we will compute the sub-Riemannian limit of the geodesic curvature of curves on surfaces in the rototranslation group. We will say that a surfaceΣ⊂ℛT,gL is regular if Σ is a Euclidean C2-smooth compact and oriented surface. In particular, we will assume that there exists a Euclidean C2-smooth function u:ℛT⟶ℝ such that(39)Σ=x1,x2,θ∈ℛT:ux1,x2,θ=0,and ux1∂x1+ux2∂x2+uθ∂θ≠0. Let ∇Hu=X1uX1+X2uX2. A point x∈Σ is called a characteristic if ∇Hux=0. We define the characteristic set(40)CΣ≔x1,x2,θ∈Σ|∇Hux1,x2,θ=0.Our computations will be local and away from characteristic points ofΣ. Let us first define p≔X1u,q≔X2u,andr≔X3˜u. We then define(41)l≔p2+q2,lL≔p2+q2+r2,p¯≔pl,q¯≔ql,pL¯≔plL,qL¯≔qlL,rL¯≔rlL.In particular,p¯2+q¯2=1. These functions are well defined at every noncharacteristic point. Let(42)vL=pL¯X1+qL¯X2+rL¯X3˜,e1=q¯X1−p¯X2,e2=rL¯p¯X1+rL¯q¯X2−llLX3˜.Then,vL is the Riemannian unit normal vector to Σ, and e1 and e2 are the orthonormal basis of Σ. On TΣ, we define a linear transformation JL:TΣ⟶TΣ such that(43)JLe1≔e2,JLe2≔−e1.For everyU,V∈TΣ, we define ∇UΣ,LV=π∇ULV, where π:TG⟶TΣ is the projection. Then, ∇Σ,L is the Levi-Civita connection on Σ with respect to the metric gL. By (27), (42), and(44)∇γ˙Σ,Lγ˙=∇γ˙Lγ˙,e1Le1+∇γ˙Lγ˙,e2Le2,we have(45)∇γ˙Σ,Lγ˙=q¯γ1..cosθ+γ2..sinθ+θ.−γ1.sinθ+γ2.cosθ+Lωγ˙t−p¯γ1.cosθ+γ2.sinθωγ˙t1−L+θ..te1+rL¯p¯γ1..cosθ+γ2..sinθ+θ.−γ1.sinθ+γ2.cosθ+Lωγ˙t+rL¯q¯γ1.cosθ+γ2.sinθωγ˙t1−L+θ..t−llLL1/2γ1.cosθ+γ2.sinθθ.−1L+ddtωγ˙te2.Moreover, ifωγ˙t=0, then(46)∇γ˙Σ,Lγ˙=q¯γ1..cosθ+γ2..sinθ+θ.−γ1.sinθ+γ2.cosθ−p¯θ..e1+rL¯p¯γ1..cosθ+γ2..sinθ+θ.−γ1.sinθ+γ2.cosθ+rL¯q¯θ−llLL1/2γ1.cosθ+γ2.sinθθ.−1L+ddtωγ˙t..e2.Definition 4.
LetΣ⊂ℛT,gL be a regular surface and γ:a,b⟶Σ be a Euclidean C2-smooth regular curve. The geodesic curvature κγ,ΣL of γ at γt is defined as(47)κγ,ΣL≔∇γ˙Σ,Lγ˙Σ,L2γ˙Σ,L4−∇γ˙Σ,Lγ˙,γ˙Σ,L2γ˙Σ,L6.Definition 5.
LetΣ⊂ℛT,gL be a regular surface and γ:a,b⟶Σ be a Euclidean C2-smooth regular curve. We define the intrinsic geodesic curvature κγ,Σ∞ of γ at γt to be(48)κγ,Σ∞≔limL⟶+∞κγ,ΣLif the limit exists.Proposition 5.
LetΣ⊂ℛT,gL be a regular surface and γ:a,b⟶Σ be a Euclidean C2-smooth regular curve. Then,(49)κγ,Σ∞=q¯θ.+p¯γ1.cosθ+γ2.sinθωγ˙tifωγ˙t≠0,κγ,Σ∞=0ifωγ˙t=0andddtωγ˙t=0,(50)limL⟶+∞κγ,ΣLL=d/dtωγ˙t−p¯θ.+q¯γ1.cosθ+γ2.sinθ2ifωγ˙t=0andddtωγ˙t≠0.Proof.
By (25) and γ˙∈TΣ, we have(51)γ˙t=γ1.cosθ+γ2.sinθX1+θ.tX2+ωγ˙tX3.
By (46), we have(52)γ˙t=ae1+be2=aq¯X1−p¯X2+brL¯p¯X1+rL¯q¯X2−llLX3˜=aq¯+brL¯p¯X1+−ap¯+brL¯q¯X2−bllLL−1/2X3.
By the aforementioned first equation and the second equation, we get(53)aq¯+brL¯p¯=γ1.cosθ+γ2.sinθ,−ap¯+brL¯q¯=θ.t,−bllLL−12=ωγ˙t.
Solving the above equations, we get(54)a=γ1.cosθ+γ2.sinθq¯−p¯θ.t,b=−L12lLlωγ˙t.
Then, we get(55)γ˙=−p¯θ.t+q¯γ1.cosθ+γ2.sinθe1−lLlL1/2ωγ˙te2.
By (26), we have(56)∇γ˙Σ,Lγ˙Σ,L2=q¯γ1..cosθ+γ2..sinθ+θ.−γ1.sinθ+γ2.cosθ+Lωγ˙t−p¯γ1.cosθ+γ2.sinθωγ˙t1−L+θ..t2+rL¯p¯γ1..cosθ+γ2..sinθ+θ.−γ1.sinθ+γ2.cosθ+Lωγ˙t+rL¯q¯γ1.cosθ+γ2.sinθωγ˙t1−L+θ..t−llLL1/2γ1.cosθ+γ2.sinθθ.−1L+ddtωγ˙t2∼q¯θ˙ωγ˙t+p¯γ˙1cosθ+γ˙2sinθωγ˙t2L2+rL¯p¯θ.ωγ˙t−rL¯q¯γ˙1cosθ+γ˙2sinθωγ˙t2L2=q¯θ.+p¯γ˙1cosθ+γ˙2sinθ2ωγ˙tL2+rL¯p¯−q¯γ˙1cosθ+γ˙2sinθ2ω2γ˙tL2.
Similarly, we have that whenωγ˙t≠0,(57)γ˙Σ,L=−p¯θ.+q¯γ1.cosθ+γ2.sinθ2+lLl2Lω2γ˙t∼L1/2ωγ˙tasL⟶+∞.
By (45) and (55), we have(58)∇γ˙Σ,Lγ˙,γ˙Σ,L=−p¯θ.+q¯γ1.cosθ+γ2.sinθ−p¯γ1.cosθ+γ2.sinθωγ˙t1−L+θ..t+q¯γ1..cosθ+γ2..sinθ+θ.−γ1.sinθ+γ2.cosθ+Lωγ˙t−llLL1/2ωγ˙trL¯p¯γ1..cosθ+γ2..sinθ+θ.−γ1.sinθ+γ2.cosθ+Lωγ˙t+rL¯q¯γ1.cosθ+γ2.sinθωγ˙t1−L+θ..t−llLL1/2γ1.cosθ+γ2.sinθθ.−1L+ddtωγ˙t∼M0L,where M0 does not depend on L. By (47), we have(59)kγ,Σ∞=limL⟶+∞kγ,ΣL=limL⟶+∞q¯θ.+p¯γ˙1cosθ+γ˙2sinθ2+rL¯2p¯−q¯γ˙1cosθ+γ˙2sinθ2ω2γ˙t=q¯θ.+p¯γ˙1cosθ+γ˙2sinθωγ˙t,if ωγ˙t≠0. When ωγ˙t=0 and d/dtωγ˙t=0, we have(60)∇γ˙Σ,Lγ˙Σ,L2=q¯γ1..cosθ+γ2..sinθ+θ.−γ1.sinθ+γ2.cosθ−p¯θ..2+rL¯p¯γ1..cosθ+γ2..sinθ+θ.−γ1.sinθ+γ2.cosθ+rL¯q¯θ..t−llLL1/2γ1.cosθ+γ2.sinθθ.−1L2∼q¯γ1..cosθ+γ2..sinθ+θ.−γ1.sinθ+γ2.cosθ−p¯θ..2asL⟶+∞,(61)γ˙Σ,L=−p¯θ.+q¯γ1.cosθ+γ2.sinθ,(62)∇γ˙Σ,Lγ˙,γ˙Σ,L=−p¯θ.+q¯γ1.cosθ+γ2.sinθq¯γ1..cosθ+γ2..sinθ+θ.−γ1.sinθ+γ2.cosθ−p¯θ..≔AB.
By (60)–(62) and (47), we get(63)κγ,Σ∞=A2B4−A2B2B6=0.
Whenωγ˙t=0 and d/dtωγ˙t≠0, we have(64)∇γ˙Σ,Lγ˙,γ˙Σ,L2∼l2lL2Lddtωγ˙t2∼Lddtωγ˙t2,γ˙Σ,L=−p¯θ.+q¯γ1.cosθ+γ2.sinθ,∇γ˙Σ,Lγ˙,γ˙Σ,L=O1,limL⟶+∞κγ,ΣLL=limL⟶+∞1LLd/dtωγ˙t2−p¯θ.+q¯γ1.cosθ+γ2.sinθ4=d/dtωγ˙t−p¯θ.+q¯γ1.cosθ+γ2.sinθ2,if ωγ˙t=0 and d/dtωγ˙t≠0, so we get (50).Definition 6.
LetΣ⊂ℛT,gL be a regular surface. Let γ:a,b⟶Σ be a Euclidean C2-smooth regular curve. The signed geodesic curvature κγ,ΣL,s of γ at γt is defined as(65)κγ,ΣL,s≔∇γ˙Σ,Lγ˙,JLγ˙Σ,Lγ˙Σ,L3,where JL is defined by (43).Definition 7.
LetΣ⊂ℛT,gL be a regular surface. Let γ:a,b⟶Σ be a Euclidean C2-smooth regular curve. We define the intrinsic geodesic curvature κγ,Σ∞ of γ at the noncharacteristic point γt to be(66)κγ,Σ∞,s≔limL⟶+∞κγ,ΣL,sif the limit exists.Proposition 6.
LetΣ⊂ℛT,gL be a regular surface. Let γ:a,b⟶Σ be a Euclidean C2-smooth regular curve. Then,(67)κγ,Σ∞,s=q¯θ.+p¯γ1.cosθ+γ2.sinθωγ˙tifωγ˙t≠0,κγ,Σ∞,s=0ifωγ˙t=0andddtωγ˙t=0,limL⟶+∞kγ,ΣL,sL=p¯θ.−q¯γ1.cosθ+γ2.sinθd/dtωγ˙t−p¯θ.+q¯γ1.cosθ+γ2.sinθ3if ωγ˙t=0 and d/dtωγ˙t≠0.Proof.
By (43) and (55), we get(68)JLγ˙=−p¯θ.+q¯γ1.cosθ+γ2.sinθJLe1−lLlL1/2ωγ˙tJLe2=lLlL1/2ωγ˙te1+−p¯θ.+q¯γ1.cosθ+γ2.sinθe2.
By (45) and the above equation, we have(69)∇γ˙Σ,Lγ˙,JLγ˙=lLlL1/2ωγ˙tq¯γ1..cosθ+γ2..sinθ+θ.−γ1.sinθ+γ2.cosθ+Lωγ˙t−p¯γ1.cosθ+γ2.sinθωγ˙t1−L+θ..t+−p¯θ.+q¯γ1.cosθ+γ2.sinθrL¯p¯γ1..cosθ+γ2..sinθ+θ.−γ1.sinθ+γ2.cosθ+Lωγ˙t+rL¯q¯γ1.cosθ+γ2.sinθωγ˙t1−L+θ..t−llLL1/2γ1.cosθ+γ2.sinθθ.−1L+ddtωγ˙t∼lLlL3/2q¯θ˙+p¯γ˙1cosθ+γ˙2sinθω2γ˙t∼L3/2ω2γ˙tq¯θ˙+p¯γ˙1cosθ+γ˙2sinθasL⟶+∞.
So, we get(70)κγ,ΣL,s=∇γ˙Σ,Lγ˙,JLγ˙Σ,Lγ˙Σ,L3=L3/2q¯θ˙+p¯γ˙1cosθ+γ˙2sinθω2γ˙tL3/2ωγ˙t3.
Furthermore,(71)κγ,Σ∞,s=limL⟶+∞kγ,ΣL,s=q¯θ˙+p¯γ˙1cosθ+γ˙2sinθωγ˙t.
Whenωγ˙t=0 and d/dtωγ˙t=0, we get(72)∇γ˙Σ,Lγ˙,JLγ˙L,Σ=−p¯θ.+q¯γ1.cosθ+γ2.sinθrL¯p¯γ1..cosθ+γ2..sinθ+θ.−γ1.sinθ+γ2.cosθ+rL¯q¯θ..−llLL1/2γ1.cosθ+γ2.sinθθ.−1L∼M0L−1/2asL⟶+∞.
So,κγ,Σ∞,s=0. When ωγ˙t=0 and d/dtωγ˙t≠0, we have(73)∇γ˙Σ,Lγ˙,JLγ˙L,Σ=−p¯θ.+q¯γ1.cosθ+γ2.sinθrL¯q¯γ1..cosθ+γ2..sinθ+θ.−γ1.sinθ+γ2.cosθ+rL¯q¯θ..−llLL1/2γ1.cosθ+γ2.sinθθ.−1L+ddtωγ˙t∼−p¯θ.+q¯γ1.cosθ+γ2.sinθ−llLddtωγ˙tL1/2asL⟶+∞,∼−−p¯θ.+q¯γ1.cosθ+γ2.sinθddtωγ˙tL1/2asL⟶+∞.
We get(74)κγ,Σ∞,s=limL⟶+∞κγ,ΣLL=limL⟶+∞−−p¯θ.+q¯γ1.cosθ+γ2.sinθd/dtωγ˙tL1/2−p¯θ.+q¯γ1.cosθ+γ2.sinθ3L=p¯θ.−q¯γ1.cosθ+γ2.sinθd/dtωγ˙t−p¯θ.+q¯γ1.cosθ+γ2.sinθ3.


## 5. The Sub-Riemannian Limit of the Riemannian Gaussian Curvature of Surfaces in the Rototranslation Group

In this section, we will compute the sub-Riemannian limit of the Riemannian Gaussian curvature of surfaces in theℛT group. We define the second fundamental form IIL of the embedding of Σ into ℛT,gL:(75)IIL=∇e1LvL,e1L∇e1LvL,e2L∇e2LvL,e1L∇e2LvL,e2L.We have the following theorem.Theorem 1.
The second fundamental formIIL of the embedding of Σ into ℛT,gL is given by(76)IIL=h11h12h21h22,where(77)h11=llLX1p¯+X2q¯−rL¯p¯q¯L,h12=h21=−lLle1,∇HrL¯L−12Lp¯L2−q¯L2−L2−rL¯22Lp¯2−q¯2,h22=−l2lL2e2,∇HLrlL+X3˜rL˜+1LpL¯qL¯rL¯+1Lp¯q¯rL¯3.Proof.
Sincee1,vLL=0ande2,vLL=0, we have(78)∇e1LvL,e1L=−∇e1Le1,vLL,∇e2LvL,e2L=−∇e2Le2,vLL.
Using the definition of the connection, identities in (5), and grouping terms, we have(79)∇e1Le1=∇q¯X1−p¯X2Lq¯X1−p¯X2=q¯X1q¯X1+q¯∇X1LX1−X1p¯X2−p¯∇X1LX2−p¯X2q¯X1+q¯∇X2LX1−X2p¯X2−p¯∇X2LX2=q¯X1q¯X1−X1p¯X2+p¯L−12LX3−p¯X2q¯X1+q¯−L+12LX3−X2p¯X2=q¯X1q¯−p¯X2q¯X1−q¯X1p¯−p¯X2p¯X2−p¯q¯L−12L−p¯q¯L+12LX3=q¯X1q¯−p¯X2q¯X1−q¯X1p¯−p¯X2p¯X2+p¯q¯LX3.
Sincep¯2+q¯2=1, we have p¯Xip¯+q¯Xiq¯=0,i=1,2,3. Thus, q¯X1q¯=−p¯X1p¯ and q¯X2q¯=−p¯X2p¯, and we have(80)∇e1Le1=−p¯X1p¯+X2q¯X1−q¯X1p¯+X2q¯X2+p¯q¯LX3.
Next, we compute the inner product of this withvL, we obtain(81)h11=−∇e1Le1,vLL=p¯p¯LX1p¯+X2q¯+q¯q¯LX1p¯+X2q¯−rL¯p¯q¯LL=plplLX1p¯+X2q¯+qlqlLX1p¯+X2q¯−rL¯p¯q¯L=1llLp2+q2X1p¯+1llLp2+q2X2q¯−rL¯p¯q¯L=llLX1p¯+X2q¯−rL¯p¯q¯L.
To computeh12 and h21, using the definition of the connection, we obtain(82)∇e1Le2=∇q¯X1−p¯X2LrL¯p¯X1+rL¯q¯X2−llLL−1/2X3=q¯∇X1LrL¯p¯X1+rL¯q¯X2−llLL−1/2X3−p¯∇X2LrL¯p¯X1+rL¯q¯X2−llLL−12X3=q¯X1rL¯p¯X1+rL¯p¯∇X1LX1+X1rL¯p¯X2+rL¯p¯∇X1LX2−X1llLL−1/2X3−llLL−1/2∇X1LX3−p¯X2rL¯p¯X1+rL¯p¯∇X2LX1+X2rL¯p¯X2+rL¯p¯∇X2LX2−X2llLL−1/2X3−llLL−1/2∇X2LX3=q¯X1rL¯p¯−p¯X2rL¯p¯+p¯llLL1+L2X1+q¯X1rL¯q¯−q¯llLL1−L2−p¯X2rL¯q¯X2+−q¯LX1llL+p¯LX2llL+rL¯21+1Lp¯2−q¯2X3.
Next, we compute the inner product of this withvL. Using the product rule and the identity qL¯p¯=pL¯q¯, we obtain(83)∇e1e2,vLL=pL¯q¯p¯+qL¯q¯2X1rL¯−pLp¯¯2+qL¯p¯q¯X2rL¯+pL¯rL¯q¯X1p¯+rL¯qL¯q¯X1q¯−rL¯p¯pL¯X2p¯+qL¯X2q¯+12Lp¯L2−q¯L2+L2+rL¯22Lp¯2−q¯2+rL¯p¯X2llL−rL¯q¯X1llL.
To simplify this, we findpL¯q¯p¯+qL¯q¯2=qL¯p¯2+qL¯q¯2=qL¯p¯2+q¯2=qL¯, pL¯p¯2+qL¯p¯q¯=pL¯p¯2+pL¯q¯2=pL¯p¯2+q¯2=pL¯, pL¯rL¯q¯X1p¯+rL¯qL¯q¯X1q¯=pL¯rL¯q¯X1p¯+rL¯qL¯−p¯X1p¯=rL¯pL¯q¯−qL¯p¯X1p¯=0, and rL¯p¯pL¯X2p¯+p¯qL¯X2q¯=rL¯pL¯p¯X2p¯+pL¯q¯X2q¯=rL¯pL¯p¯X2p¯+q¯X2q¯=0. Under these simplifications, we get(84)∇e1e2,vLL=qL¯X1rL¯−pL¯X2rL¯+12Lp¯L2−q¯L2+L2+rL¯22Lp¯2−q¯2+rL¯p¯X2llL−rL¯q¯X1llL=llLe1,∇HrL¯L−rL¯e1,∇HllLL+12Lp¯L2−q¯L2+L2+rL¯22Lp¯2−q¯2.
Finally, we use the identityl/lL−lL/l∇HrL¯=rL¯∇Hl/lL in the above equation:(85)∇e1e2,vLL=lLle1,∇HrL¯L+12Lp¯L2−q¯L2+L2+rL¯22Lp¯2−q¯2.
Therefore,(86)h12=h21=−∇e1e2,vLL=−lLle1,∇HrL¯L−12Lp¯L2−q¯L2−L2−rL¯22Lp¯2−q¯2.
Since∇e2vL,e2L=−∇e2e2,vLL, using the definition of connection, identities in (5), and grouping terms, we have(87)∇e2Le2=∇rL¯p¯X1+rL¯q¯X2−l/lLL−1/2X3LrL¯p¯X1+rL¯q¯X2−llLL−1/2X3=rL¯p¯X1rL¯p¯+rL¯q¯X2rL¯p¯−lLlLX3rL¯p¯−lLlLrL¯q¯X1+rL¯p¯X1rL¯q¯+rL¯q¯X2rL¯q¯−lLlLX3rL¯q¯−lrL¯p¯LlL1−LX2+−rL¯p¯LX1llL−rL¯q¯LX2llL+llLLX3llL−rL¯2p¯q¯LX3.
Taking the inner product withvL yields(88)∇e2Le2,vLL=pL¯rL¯p¯2X1rL¯+pL¯rL¯2p¯X1p¯+pL¯rL¯q¯p¯X2rL¯+pL¯rL¯2q¯X2p¯−pL¯lLlLp¯X3rL¯−pL¯lLlLrL¯X3p¯−pL¯lLlLrL¯q¯+qL¯rL¯p¯q¯X1rL¯+qL¯rL¯2p¯X1q¯+qL¯rL¯2q¯X2rL¯+qL¯rL¯2q¯X2q¯−lq¯qL¯lLLX3rL¯−q¯lrL¯lLLX3q¯−qL¯lrL¯p¯lLL1−L+rL¯L−p¯rL¯LX1llL−q¯rL¯LX2llL+llLLX3llL−rL¯2p¯q¯L.
Under some similar simplifications to Theorem 4.3 in [8], we get(89)h22=−∇e2Le2,vLL=−l2lL2e2,∇HLrlL+X3˜rL¯+1LpL¯qL¯rL¯+1Lp¯q¯rL¯3.
The Riemannian mean curvatureℋL of Σ is defined by(90)ℋL≔trIIL=llLX1p¯+X2q¯−rL¯p¯q¯L−l2lL2e2,∇HLrlL+X3˜rL¯+1LpL¯qL¯rL¯+1Lp¯q¯rL¯3.
Similar to Proposition 3.8 in [15], away from the characteristic point, the horizontal mean curvature ℋ∞ of Σ∈ℛT,gL is given by(91)ℋ∞=limL⟶∞ℋL=X1p¯+X2q¯.
Let(92)KΣ,Le1,e2=−RΣ,Le1,e2e1,e2Σ,L,KLe1,e2=−RLe1,e2e1,e2L.
By the Gauss equation, we have(93)KΣ,Le1,e2=KLe1,e2+detIIL.Proposition 7.
Away from characteristic points, we have(94)KΣ,∞e1,e2=−e1,∇HX3u∇Hu−X3u2l2asL⟶+∞.Proof.
We compute(95)RLe1,e2e1=RLq¯X1−p¯X2,r¯Lp¯X1+r¯Lq¯X2−llLLX3q¯X1−p¯X2=r¯Lp¯q¯2RLX1,X1X1+r¯Lq¯3RLX1,X2X1−lq¯2lLLRLX1,X3X1−r¯Lp¯2q¯RLX2,X1X1−r¯Lp¯q¯2RLX2,X2X1+lp¯q¯lLLRLX2,X3X1−r¯Lp¯2q¯RLX1,X1X2−r¯Lp¯q¯2RLX1,X2X2+lp¯q¯lLLRLX1,X3X2+r¯Lp¯3RLX2,X1X2+r¯Lp¯2q¯RLX2,X2X2−lp¯2lLLRLX2,X3X2=rL¯q¯RLX1,X2X1−lq¯lLLRLX1,X3X1+lp¯q¯lLLRLX2,X3X1−rL¯p¯RLX1,X2X2+lp¯q¯lLLRLX1,X3X2−lp¯2lLLRLX2,X3X2=−3L2+2L+14LrL¯p¯X1+rL¯q¯3L2−2L−14LX2+p¯2llLL−L2−2L−14L+lp¯2lLL−L2−2L+34LX3,KLe1,e2=−RLe1,e2e1,e2L=−34L+12+14LrL¯2p¯2−rL¯2q¯234L−12−14L−p¯2llL2−L4−12+34L−l2q¯2lL2−L4+12−14L.
To simplify this, we find−3/4LrL¯2p¯2−3/4LrL¯2q¯2=−3/4LrL¯2∼−3/4X3u2/l2asL⟶∞, p¯2l/lL2L/4+q¯2l/lL2L/4=l/lL2L/4, and p¯2l/lL21/2−q¯2l/lL21/2=1/2−q¯2l/lL2,l/lL2∼1asL⟶∞. Finally, we get(96)KLe1,e2∼−34X3u2l2+L4llL2+12−q¯2asL⟶∞.
By Theorem1 and ∇Hr¯L=L−1/2∇HX3u/∇Hu+OL−1 as L⟶+∞, we get(97)detIIL=h11h22−h122=−L4−e1,∇HX3u∇Hu+12q¯2−p¯2+OL−1/2as L⟶+∞. Here, we have used the equation L/4l/lL2−L/4=−L/4lL2−l2/lL2=−L/4r2/lL2=−L/4lL2X3u2/L=−X3u2/4lL2∼−X3u2/4l2 as L⟶+∞. By (93), (96), and (97), we get the desired equation. □


## 6. A Gauss–Bonnet Theorem in the Rototranslation Group

In this section, we will prove the Gauss–Bonnet theorem in the rototranslation group. Firstly, we consider the case of a regular curveγ:a,b⟶ℛT,gL. We define the Riemannian length measure(98)dsL=γ˙Ldt.Lemma 1.
Letγ:a,b⟶ℛT,gL be a Euclidean C2-smooth and regular curve. Let(99)ds≔ωγ˙tdt,ds¯≔121ωγ˙tγ˙1tcosθ+γ˙2tsinθ2+θ˙t2dt.
Then,(100)limL⟶∞1L∫γdsL=∫abds.
Whenωγ˙t≠0, we have(101)1LdsL=ds+ds¯L−1+OL−2asL⟶+∞.
Whenωγ˙t=0, we have(102)1LdsL=1Lγ˙1tcosθ+γ˙2tsinθ2+θ˙t2dt.Proof.
Since(103)γ˙tL=γ˙1tcosθ+γ˙2tsinθ2+θ˙t2+Lωγ˙t2,similar to the proof of Lemma 6.1 in [5], we can prove(104)limL⟶∞1L∫γγ˙tLdt=∫ablimL⟶∞1Lγ˙tLdt=∫ablimL⟶∞1Lγ˙1tcosθ+γ˙2tsinθ2+θ˙t2+Lωγ˙t2dt=∫abωγ˙tdt=∫abds.
Whenωγ˙t≠0, we have(105)1LdsL=L−1γ˙1tcosθ+γ˙2tsinθ2+θ˙t2+ωγ˙t2dt.
Using the Taylor expansion, we can prove(106)1LdsL=ds+ds¯L−1+OL−2asL⟶+∞.
From the definition ofdsL and ωγ˙t=0, we get(107)1LdsL=1Lγ˙1tcosθ+γ˙2tsinθ2+θ˙t2dt.Proposition 8.
LetΣ⊂ℛT,gL be a Euclidean C2-smooth surface, Σ=u=0, and dσΣ,L denote the surface measure on Σ with respect to the Riemannian metric gL. Let(108)dσΣ≔p¯ω2−q¯ω1∧ω,dσ¯Σ≔X3ulω1∧ω2−X3u22l2p¯ω2−q¯ω1∧ω.
Then,(109)1LdσΣ,L=dσΣ+dσ¯ΣL−1+OL−2,asL⟶+∞.
IfΣ=fD with f=fu1,u2=f1,f2,f3:D⊂ℝ2⟶ℛT,gL, then(110)limL⟶∞1L∫ΣdσΣ,L=∫Df1u2f3u1−f1u1f3u2sinθ+f2u1f3u2−f2u2f3u1cosθ2+f1u1f2u2−f1u2f2u121/2du1du2.Proof.
It is well known that(111)gLX1,⋅=ω1,gLX2,⋅=ω2,gLX3,⋅=Lω.
We definee1∗≔gLe1,⋅ande2∗≔gLe2,⋅; then,(112)e1∗=q¯ω1−p¯ω2,e2∗=r¯Lp¯ω1+r¯Lq¯ω2−llLL1/2ω.
Therefore,(113)1LdσΣ,L=1Le1∗∧e2∗=llLp¯ω2−q¯ω1∧ω+1Lr¯Lω1∧ω2.
Recalling(114)r¯L=X3uL−1/2p2+q2+L−1X3u2and the Taylor expansion(115)1lL=1l−12l3X3u2L−1+OL−2asL⟶+∞,we get (109). By (5), we have(116)fu1=f1u1∂x1+f2u1∂x2+f3u1∂θ=f1u1cosX1+sinX3+f2u1sinX1−cosX3+f3u1X2=f1u1cosθ+f2u1sinθX1+f3u1X2+Lf1u1sinθ−f2u1cosθX3˜,fu2=f1u2∂x1+f2u2∂x2+f3u2∂θ=f1u2cosθ+f2u2sinθX1+f3u2X2+Lf1u2sinθ−f2u2cosθX3˜.
Let(117)v¯L=X1X2X3˜f1u1cosθ+f2u1sinθf3u1Lf1u1sinθ−f2u1cosθf1u2cosθ+f2u2sinθf3u2Lf1u2sinθ−f2u2cosθ,=Lf1u2f3u1−f1u1f3u2sinθ+f2u1f3u2−f2u2f3u1cosθX1+Lf1u1f2u2−f1u2f2u1X2+f1u1f3u2−f1u2f3u1cosθ+f2u1f3u2−f2u2f3u1sinθX3˜.
We know thatdσΣ,L=detgijdu1du2,gij=gLfui,fuj, and(118)detgij=v¯LL2=Lf1u2f3u1−f1u1f3u2sinθ+f2u1f3u2−f2u2f3u1cosθ2+Lf1u1f2u2−f1u2f2u12+f1u1f3u2−f1u2f3u1cosθ+f2u1f3u2−f2u2f3u1sinθ2,so by the dominated convergence theorem, we get(119)limL⟶∞1L∫ΣdσΣ,L=∫Df1u2f3u1−f1u1f3u2sinθ+f2u1f3u2−f2u2f3u1cosθ2+f1u1f2u2−f1u2f2u121/2du1du2.
Similar to the proof of Theorem 4.3 in [6], we get a Gauss–Bonnet theorem in the rototranslation group as follows.Theorem 2.
LetΣ⊂ℛT,gL be a regular surface with finitely many boundary components ∂Σi,i∈1,…,n, given by Euclidean C2-smooth regular and closed curves γi:0,2π⟶∂Σi. Suppose that the characteristic set CΣ satisfies ℋ1CΣ=0 where ℋ1CΣ denotes the Euclidean 1-dimensional Hausdorff measure of CΣ and that ∇HuH−1 is locally summable with respect to the Euclidean 2-dimensional Hausdorff measure near the characteristic set CΣ; then,(120)∫ΣKΣ,∞dσΣ+∑i=1n∫γiκγi,Σ∞,sds=0.Proof.
Using the similar discussions in [13, 14], we assume that all points satisfy ωγ˙it≠0 on the curve γi. Recalling the result in Proposition 4 indicates(121)κγi,ΣL,s=κγi,Σ∞,s+OL−1/2.
According to the Gauss–Bonnet theorem, we get(122)∫ΣKΣ,L1LdσΣ,L+∑i=1n∫γiκγi,ΣL,s1LdsL=2πχΣL.
LetL go to the infinity, and using the dominated convergence theorem, (121), (122), (109), Proposition 6, and Lemma 1, we get the desired result.


## 7. Conclusion

This paper dealt with an interesting question of the Gauss-Bonnet theorem in the rototranslation group from the Riemannian approximation scheme. The main result of this paper is Theorem2, which is the Gauss–Bonnet-type theorem in the rototranslation group. To prove Theorem 2, we obtained the sub-Riemannian limit of the curvature of curves, sub-Riemannian limits of the geodesic curvature of curves on surfaces, and the Riemannian Gaussian curvature of surfaces in the rototranslation group.As a future work, we plan to proceed to study Gauss–Bonnet theorems in the rototranslation group with the general left-invariant metric and other three-dimensional Riemannian Lie groups which were classified in [18]. In these conditions, Gauss–Bonnet theorems can be obtained through the Riemannian approximation scheme took by Balogh et al. [13, 14]. The 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 the Gauss–Bonnet theorem to other different Lie groups. We believe that the results to be obtained will have some geometric applications.



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