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

&e 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.


Introduction
e 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]. eir 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. is approach showed that Hawking radiation possesses a topological effect coming from the Euler characteristic of the spacetime. e 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].
ey 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. e 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 R 2 × S 1 with coordinates (x, y, θ). e 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. e 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 positionorientation 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 C 2 -smooth surface in the Heisenberg group H 1 away from characteristic points and a notion of the intrinsic signed geodesic curvature for Euclidean C 2 -smooth curves on surfaces. ese results were then used to prove a Heisenberg version of the Gauss-Bonnet theorem. ey proposed an interesting question to understand to what extent similar phenomena hold in other sub-Riemannian geometric structures. In [15][16][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. eir approach is to define sub-Riemannian objects as limits of horizontal objects in (G, g L ), where a family of metrics g L is essentially obtained as an anisotropic blowup of the Riemannian metric g 1 . 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 C 2 -smooth surface in the rototranslation group away from characteristic points and signed geodesic curvature for Euclidean C 2 -smooth curves on surfaces. We also obtain a Gauss-Bonnet theorem in the rototranslation group.
In Section 2, 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.

Proposition 1.
Let RT be the rototranslation group, relative to the coordinate frame X 1 , X 2 , X 3 ; then, the Levi-Civita connection on RT is given by Proof. It follows from a direct application of the Koszul identity, which here simplifies where i, j, k � 1, 2, 3. By (3) and (6), we have When j � 1, we compute 〈∇ L we get ∇ L X 1 X 2 � (L − 1/2L)X 3 . Other cases follow the similar way.
□ Define the curvature of the connection ∇ L by We get the following proposition.

Proposition 2.
Let RT be the rototranslation group; then, Proof. It is a direct computation using Taking for example, we compute Hence, We compute the sectional curvatures of the two planes spanned by the basis vectors X i and X j : In fact, the full Riemannian curvature tensor In order to compute the Kretschmann scalar, from R ijkl L � g ll g kk g jj g ii R L ijkl , it follows that we can write if (ijkl) � (1313), (3131), (2323) or (3232),

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Recall that the Kretschmann scalar is defined by Next, the Ricci curvature while the scalar curvature It can be observed that the Kretschmann scalar and the sectional, Ricci, and scalar curvatures all diverge as L ⟶ ∞.

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.

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In particular, if c(t) is a horizontal point of c, By Proposition 1 and (25), we have By (22) and (25), we get By the definition of κ L c , we get Proposition 3.
if the limit exists.
We introduce the following notation: for continuous erefore, By (24) and When If

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 surface Σ ⊂ (RT, g L ) 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: RT ⟶ R such that and Our computations will be local and away from characteristic points of Σ. Let us first define p ≔ X 1 u, q ≔ X 2 u, and r ≔ X 3 u. We then define In particular, p 2 + q 2 � 1. ese functions are well defined at every noncharacteristic point. Let v L � p L X 1 + q L X 2 + r L X 3 , en, v L is the Riemannian unit normal vector to Σ, and e 1 and 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 � π∇ L U V, where π: TG ⟶ TΣ is the projection. en, ∇ Σ,L is the Levi-Civita connection on Σ with respect to the metric g L . By (27), (42), and we have Journal of Mathematics Definition 5. Let Σ ⊂ (RT, g L ) be a regular surface and c: [a, b] ⟶ Σ be a Euclidean C 2 -smooth regular curve. We define the intrinsic geodesic curvature if the limit exists.
if the limit exists.

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 the RT group. We define the second fundamental form II L of the embedding of Σ into (RT, g L ): We have the following theorem.

Theorem 1.
e second fundamental form II L of the embedding of Σ into (RT, g L ) is given by where

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Proof. Since 〈e 1 , v L 〉 L � 0 and 〈e 2 , v L 〉 L � 0, we have Using the definition of the connection, identities in (5), and grouping terms, we have Since p 2 + q 2 � 1, we have pX i p + qX i q � 0, i � 1, 2, 3. us, qX 1 q � − pX 1 p and qX 2 q � − pX 2 p, 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

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Next, we compute the inner product of this with v L . Using the product rule and the identity q L p � p L q, we obtain 〈∇ e 1 e 2 , v L 〉 L � p L qp + q L q 2 X 1 r L − p L p 2 + q L pq X 2 r L + p L r L qX 1 p + r L q L qX 1 q To simplify this, we find p L qp + q L q 2 � (q L p 2 + q L q 2 ) � q L (p 2 + q 2 ) � q L , p L p 2 + q L pq � p L p 2 + p L q 2 � p L (p 2 +q 2 ) � p L , p L r L qX 1 p + r L q L qX 1 q � p L r L qX 1 p + r L q L (− pX 1 p) � r L (p L q − q L p)X 1 p � 0, and r L (pp L X 2 p +pq L X 2 q) � r L (p L pX 2 p + p L qX 2 q) � r L p L (pX 2 p + qX 2 q) � 0. Under these simplifications, we get Finally, we use the identity ((l/l L ) − (l L /l))∇ H r L � r L ∇ H (l/l L ) in the above equation: erefore, Since 〈∇ e 2 v L , e 2 〉 L � − 〈∇ e 2 e 2 , v L 〉 L , using the definition of connection, identities in (5), and grouping terms, we have Taking the inner product with v L yields 16 Journal of Mathematics Under some similar simplifications to eorem 4.3 in [8], we get e Riemannian mean curvature H L of Σ is defined by (90) Similar to Proposition 3.8 in [15], away from the characteristic point, the horizontal mean curvature H ∞ of Σ ∈ (RT, g L ) is given by Proof. We compute

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 curve c: [a, b] ⟶ (RT, g L ). We define the Riemannian length measure Then, 18 Journal of Mathematics (100) When ω( _ c(t)) ≠ 0, we have When ω( _ c(t)) � 0, we have similar to the proof of Lemma 6.1 in [5], we can prove When ω( _ c(t)) ≠ 0, we have Using the Taylor expansion, we can prove From the definition of ds L and ω( _ c(t)) � 0, we get Then, If (110)