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

Abstract

The rototranslation group 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 -smooth surface in the rototranslation group away from characteristic points and signed geodesic curvature for Euclidean -smooth curves on surfaces. Based on these results, we obtain a Gauss–Bonnet theorem in the rototranslation group.

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, , is the group of Euclidean rotations and translations of the plane equipped with a particular sub-Riemannian metric. More precisely, is a three-dimensional topological manifold diffeomorphic to with coordinates . 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 and its universal cover were introduced. The main theorem states that a straight ruled surface in 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 -smooth surface in the Heisenberg group away from characteristic points and a notion of the intrinsic signed geodesic curvature for Euclidean -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 , where a family of metrics is essentially obtained as an anisotropic blowup of the Riemannian metric . At the heart of this approach is the fact that the intrinsic horizontal geometry does not change with . 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 -smooth surface in the rototranslation group away from characteristic points and signed geodesic curvature for Euclidean -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.

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 group is the group comprising rotations and translations of the Euclidean plane. It is a 3-dimensional Lie group, isomorphic to with multiplication given byfor all . In this model, the natural element of is , and the inverse element of is .

Letwith brackets

Then,and . Let be the horizontal distribution on . Let . Then, . To describe the Riemannian approximants to , for the constant , we define a metric so that are orthonormal bases on with respect to . It is easy to check that is the Riemannian metric on .

To compute the sectional, Ricci, and scalar curvatures of the rototranslation group with respect to , we use the Levi-Civita connection on . A straightforward calculation shows the following proposition.

Proposition 1. Let be the rototranslation group, relative to the coordinate frame ; then, the Levi-Civita connection on is given by

Proof. It follows from a direct application of the Koszul identity, which here simplifieswhere . By (3) and (6), we haveWhen , we compute . It follows that , , and Hence, . Similarly, and . By the following equation,we get . Other cases follow the similar way.

Define the curvature of the connection by

We get the following proposition.

Proposition 2. Let be the rototranslation group; then,

Proof. It is a direct computation usingTakingfor example, we computeHence,We compute the sectional curvatures of the two planes spanned by the basis vectors and : , where for and .In fact, the full Riemannian curvature tensor isIn order to compute the Kretschmann scalar, from , it follows that we can writeRecall that the Kretschmann scalar is defined by . We calculate thatNext, the Ricci curvature iswhile the scalar curvatureIt can be observed that the Kretschmann scalar and the sectional, Ricci, and scalar curvatures all diverge as .

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. Let be a Euclidean -smooth curve; we say that is regular if for every . Moreover, we say that is a horizontal point of ifwhere .

Definition 2. Let be a Euclidean -smooth regular curve in the Riemannian manifold . The curvature of at is defined as

Proposition 3. Let be a Euclidean -smooth regular curve in the Riemannian manifold . Then,In particular, if is a horizontal point of ,

Proof. By (4), we haveBy Proposition 1 and (25), we haveBy (25) and (26), we haveBy (22) and (25), we getBy the definition of , we get Proposition 3.

Definition 3. Let be a Euclidean -smooth regular curve in the Riemannian manifold ; we define the intrinsic curvature of at to beif the limit exists.
We introduce the following notation: for continuous functions ,

Proposition 4. Let be a Euclidean -smooth regular curve in the Riemannian manifold . Then,

Proof. When , we haveTherefore,If , by (22), we haveBy (24) and , we haveWhen and , we haveIf and by (22), we get

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 surface is regular if is a Euclidean -smooth compact and oriented surface. In particular, we will assume that there exists a Euclidean -smooth function such thatand . Let . A point is called a characteristic if . We define the characteristic set

Our computations will be local and away from characteristic points of . Let us first define . We then define

In particular, . These functions are well defined at every noncharacteristic point. Let

Then, is the Riemannian unit normal vector to , and and are the orthonormal basis of . On , we define a linear transformation such that

For every , we define , where is the projection. Then, is the Levi-Civita connection on with respect to the metric . By (27), (42), andwe have

Moreover, if , then

Definition 4. Let be a regular surface and be a Euclidean -smooth regular curve. The geodesic curvature of at is defined as

Definition 5. Let be a regular surface and be a Euclidean -smooth regular curve. We define the intrinsic geodesic curvature of at to beif the limit exists.

Proposition 5. Let be a regular surface and be a Euclidean -smooth regular curve. Then,

Proof. By (25) and , we haveBy (46), we haveBy the aforementioned first equation and the second equation, we getSolving the above equations, we getThen, we getBy (26), we haveSimilarly, we have that when ,By (45) and (55), we havewhere does not depend on . By (47), we haveif . When and , we haveBy (60)–(62) and (47), we getWhen and , we haveif and , so we get (50).

Definition 6. Let be a regular surface. Let be a Euclidean -smooth regular curve. The signed geodesic curvature of at is defined aswhere is defined by (43).

Definition 7. Let be a regular surface. Let be a Euclidean -smooth regular curve. We define the intrinsic geodesic curvature of at the noncharacteristic point to beif the limit exists.

Proposition 6. Let be a regular surface. Let be a Euclidean -smooth regular curve. Then,if and .

Proof. By (43) and (55), we getBy (45) and the above equation, we haveSo, we getFurthermore,When and , we getSo, . When and , we haveWe get

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 the group. We define the second fundamental form of the embedding of into :

We have the following theorem.

Theorem 1. The second fundamental form of the embedding of into is given bywhere

Proof. Since , we haveUsing the definition of the connection, identities in (5), and grouping terms, we haveSince , we have . Thus, and , and we haveNext, we compute the inner product of this with , we obtainTo compute and , using the definition of the connection, we obtainNext, we compute the inner product of this with . Using the product rule and the identity , we obtainTo simplify this, we find , , and Under these simplifications, we getFinally, we use the identity in the above equation:Therefore,Since , using the definition of connection, identities in (5), and grouping terms, we haveTaking the inner product with yieldsUnder some similar simplifications to Theorem 4.3 in [8], we getThe Riemannian mean curvature of is defined bySimilar to Proposition 3.8 in [15], away from the characteristic point, the horizontal mean curvature of is given byLetBy the Gauss equation, we have

Proposition 7. Away from characteristic points, we have

Proof. We computeTo simplify this, we find , , and . Finally, we getBy Theorem 1 and as , we getas . Here, we have used the equation as . 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 curve . We define the Riemannian length measure

Lemma 1. Let be a Euclidean -smooth and regular curve. LetThen,When , we haveWhen , we have

Proof. Sincesimilar to the proof of Lemma 6.1 in [5], we can proveWhen , we haveUsing the Taylor expansion, we can proveFrom the definition of and , we get

Proposition 8. Let be a Euclidean -smooth surface, , and denote the surface measure on with respect to the Riemannian metric . LetThen,If with , then

Proof. It is well known thatWe define ; then,Therefore,Recallingand the Taylor expansionwe get (109). By (5), we haveLetWe know that , andso by the dominated convergence theorem, we getSimilar to the proof of Theorem 4.3 in [6], we get a Gauss–Bonnet theorem in the rototranslation group as follows.

Theorem 2. Let be a regular surface with finitely many boundary components , given by Euclidean -smooth regular and closed curves . Suppose that the characteristic set satisfies where denotes the Euclidean 1-dimensional Hausdorff measure of and that is locally summable with respect to the Euclidean 2-dimensional Hausdorff measure near the characteristic set ; then,

Proof. Using the similar discussions in [13, 14], we assume that all points satisfy on the curve . Recalling the result in Proposition 4 indicatesAccording to the Gauss–Bonnet theorem, we getLet 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 Theorem 2, 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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest in this work.

Authors’ Contributions

All the authors contributed equally to the writing of this paper and read and approved the final manuscript.

Acknowledgments

This research was funded by the Reform and Development Foundation for Local Colleges and Universities of the Central Government (Excellent Young Talents Project of Heilongjiang Province, Grant no. ZYQN2019071) and the Project of Science and Technology of Heilongjiang Provincial Education Department (Grant nos. 1355MSYYB005 and 1354ZD008).