Sub-Lorentzian Geometry of Curves and Surfaces in a Lorentzian Lie Group

Liu, Haiming; Guan, Jianyun

doi:https://doi.org/10.1155/2022/5396981

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Volume 2022 | Article ID 5396981 | https://doi.org/10.1155/2022/5396981

Sub-Lorentzian Geometry of Curves and Surfaces in a Lorentzian Lie Group

Haiming Liu¹and Jianyun Guan¹

Academic Editor: Claudio Dappiaggi

Received21 Mar 2022

Accepted14 May 2022

Published02 Jun 2022

Abstract

We consider the sub-Lorentzian geometry of curves and surfaces in the Lie group Firstly, as an application of Riemannian approximants scheme, we give the definition of Lorentzian approximants scheme for which is a sequence of Lorentzian manifolds denoted by . By using the Koszul formula, we calculate the expressions of Levi-Civita connection and curvature tensor in the Lorentzian approximants of in terms of the basis 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 points. Furthermore, we derive the expressions of those curvatures and prove two generalized Gauss-Bonnet theorems in .

1. Introduction

In recent years, there has been done a lot of research concerning Gauss-Bonnet theorems in three-dimensional Lie groups with sub-Riemannian geometric structures. One of the reasons for this success is the discovery of the method of Riemannian approximations. Initial work by Balogh et al. proved a Heisenberg version of the Gauss-Bonnet theorem with the help of the method of Riemannian approximations [1, 2]. Hereafter the above work, many research on Gauss-Bonnet theorem have been gained; we refer the monograph [3–6]. In particular, Wang and Wei proved Gauss-Bonnet theorems on the affine group [3], the group of rigid motions of the Minkowski plane [3, 4], the BCV spaces [5], and the Lorentzian Heisenberg group [4, 6]. Inspired by their work, we proved Gauss-Bonnet theorems in the rototranslation group [7, 8], Lorentzian Sasakian space forms, and the group of rigid motions of Minkowski plane with the general left-invariant metric [9, 10]. Riemannian approximations can be extended to the case for any Lie group equipped with left-invariant Lorentzian metric named Lorentzian approximations. In particular, one can consider a sequence of Lorentzian manifolds denoted by , where a family of metrics , for , is essentially obtained as an anisotropic blow-up of the Lorentzian metric Then, one can define sub-Lorentzian objects as limits of horizontal objects in , since the intrinsic horizontal geometry does not change with Some typical works of Lorentzian approximations in the Lorentzian Heisenberg group are obtained in [4, 6]. For Lorentzian Sasakian space forms, see [9].

In this paper, we consider sub-Lorentzian geometry of curves and surfaces on . The group of rigid motions of Minkowski plane has two left-invariant Lorentzian metrics and [11, 12]. Onda proved that the metric is a Lorentz Ricci soliton [12]. In [13], Patrangenaru proved that any left-invariant metric on is isometric to one of the metric with and In [14], the metric was denoted by with the metric provides a natural -parametric deformation family of which is the model space of solve geometry in the eight model geometries of Thurston, which makes very interesting and important [14, 15]. However, very little is known about sub-Lorentzian geometry of In this paper, we focus on the general left-invariant Lorentzian metric on , where the coframe By using the method of Lorentzian approximations, we consider a sequence of Lorentzian manifolds , where , for , is an anisotropic blow-up of the Lorentzian metric Using the Koszul formula, we calculate the expressions of Levi-Civita connection and curvature tensor in the Lorentzian approximants of in terms of the basis We define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on spacelike surfaces and Lorentzian surfaces, and the intrinsic Gaussian curvature of spacelike surfaces and Lorentzian surfaces away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove two generalized Gauss-Bonnet theorems in .

This paper is organized as follows. In Section 2, we introduce the Lorentzian approximations of and calculate the expressions of corresponding Levi-Civita connection in terms of the basis . Furthermore, we define the notions of geodesic curvature and intrinsic geodesic curvature of curves in . We get the expressions of those curvatures and give an example. In Sections 3 and 4, we compute intrinsic geodesic curvatures of regular curves on Lorentzian surfaces and the intrinsic Gaussian curvature of Lorentzian surfaces in We also give two examples. In Section 5, we get the first Gauss-Bonnet theorem in In Section 6, we compute intrinsic geodesic curvature of curves on spacelike surfaces and the intrinsic Gaussian curvature of spacelike surfaces in and we get the second Guass-Bonnet theorem.

2. Curvatures for Curves in Lorentzian Approximations

In this section, some basic notions in the Lorentzian group of rigid motions of the Minkowski plane group will be introduced. Let be the motion group of the Minkowski -space. This consists of all matrices of the form

Topologically, is a Lie group which is diffeomorphic to Its Lie algebra has a basis consisting of for which

From the above equations in Equation (2), we get that

We denote Let be the horizontal distribution on and

Then, . We consider the left-invariant Lorentzian metrics given by and its anisotropic blow-up , for We call the Lorentzian approximants of and denote by throughout the paper. It is easy to check that is the Lorentzian metric on and are pseudo-orthonormal basis on with respect to Then, the Levi-Civita connection of is given by following proposition.

Proposition 1. The Levi-Civita connection relative to the coordinate frame of is described as follows:

Proof. To derive the above expressions, it is useful to recall the following Koszul identity from the famous proof of the unique of Riemannian connection: where By Equations (3) and (14), the first formula in Equation (11) is as follows: When , we compute . It follows that and Hence, . Similarly, and Moreover, other expressions follow some similar computations.
A parametrized curve in is a map , where is an open interval in , where and each function has derivatives of two orders, for all . Such is called -smooth. The regular curve is called regular provided that there does not exist with We call a spacelike curve, timelike curve, or null curve if is a spacelike vector, timelike vector, or null vector at any , respectively.

Definition 2. For an arbitrary -smooth regular curve , , we say that is a horizontal point of provided the following function satisfies the following:

Definition 3. For an arbitrary -smooth regular curve , , we define the geodesic curvature of at in the following way. if is a spacelike vector. if is a timelike vector.

Proposition 4. There are formulae of curvature for -smooth regular curve (1)If is a spacelike vector, then where In particular, if is a horizontal point of , (2)If is a timelike vector, then In particular, if is a horizontal point of ,

Proof. By Equation (8), we have the following: By Proposition 1 and Equation (25), we get the following: Using Equations (25) and (26), we have the following: By Equations (14), (25), and (29), we obtain the following:

By the definition of , we get the desired formulae of curvature.

Definition 5. Let be a -smooth regular curve, we define the intrinsic curvature of at to be if the limit exists.
To derive the expression of intrinsic curvature, we need the following notion: for continuous functions ,

Proposition 6. Let be a -smooth regular curve in . (1)If is a spacelike vector and , then we have the following formula of If , thenIf and , then If and , then (2)If is a timelike vector and , then we have the following formula of If , thenIf and , then

Proof. (1)If is a spacelike vector, when , we have the following:Therefore, If , by Equation (7), we have the following: By Equation (20) and , we have the following: When and , we have the following: If and , by (Equation (15)), we get the following: (2)By some similar computations, we get (2).

Example 7. Let be a -smooth regular curve, where is an open interval in and . We compute the following: By Equation (29), we have the following: Suppose that be a -smooth regular curve, and Then, we have the following: Therefore, we have the following: It implies that is a timelike vector and By Proposition 6, we have the following: We can use the above example to illustrate the meaning of Definition 5. We assume that Then, we have the following: Therefore, we have the following: It implies that is a timelike vector and By Proposition 6, we have the following:

3. Geodesic Curvatures for Curves on Lorentzian Surfaces in Lorentzian Approximations

In this section, we will compute geodesic curvatures for curves on Lorentzian surfaces in Lorentzian approximations . We consider a regular surface such that is a -smooth compact and oriented surface. Supposed that where is a -smooth function and . A point is called characteristic if where . We call the set characteristic set of . We will give some symbols away from characteristic points of . We define , , and If , we say that is a horizontal spacelike surface. Under this assumption, for Therefore, we can define the following functions: which will be used to define a frame of . We construct the following: where is the unit spacelike normal vector to , is the unit timelike vector, and is the unit spacelike vector. One can check that are a pseudo-orthonomal basis for tangent space of We call a Lorentzian surface in . For the case that is a -smooth timelike curve and , we define the following:

For the case that is a -smooth spacelike curve, we define the following:

That makes and have the same orientation with . For every , we define the Levi-Civita connection with respect to the metric on by , where is the projection. Furthermore,

In particular, we get the following:

Moreover, if , then

Definition 8. Let be a -smooth regular curve, where is a regular Lorentzian surface. We define the geodesic curvature as follows. (1)If is spacelike vector, the geodesic curvature of at is defined by the following: (2)If is timelike vector, the geodesic curvature of at is defined by the following:

Definition 9. Let be a -smooth regular curve, where is a regular Lorentzian surface. We define the intrinsic geodesic curvature of at to be if the limit exists.

Proposition 10. Let be a -smooth regular curve, where is a regular Lorentzian surface. (1)If is a timelike vector, then (2)If is a spacelike vector, then

Proof. On the one hand, by Equation (25), we get the following: On the other hand, since , we denote the following: The above equations yield the following: By solving Equation (75), we get the following: Therefore, takes the following form: We denote the following: Then, by Equation (64), we get , where In case , we obtain the following: By Equations (64) and (77), we obtain the following: where and does not depend on . By Equation (64), we have the following: If and , we have the following: where , and By Equations (79)–(86) and Equation (65), we get the following: When , and , we have the following: It follows that if , and . This completes the proof.

Example 11. We assume that there exists a -smooth function such that Then, . Let So, we have the following: Therefore, , so is a horizontal spacelike surface. By Equation (53), we have the following: By Equation (60), we have the following: Then, Thus, it is concluded that is a Lorentzian surface in
Let be the circle centered at the origin on By and , we have the following: By Equation (64), we have the following: Then, If , then In this case, Then, we have is a timelike vector. By Proposition 10 and Equation (69), we have at the point which satisfies : When we assume that be a circle centered at the origin on We get the following: By Equation (64), we have the following: So, If , then In this case, Then, we have is a timelike vector. By Proposition 10 and Equation (69), we have at the point which satisfies

Definition 12. Let be a -smooth regular curve, where is a regular Lorentzian surface. The signed geodesic curvature of at is defined as follows: where is defined by Equations (61) and (62).

Definition 13. Let be a -smooth regular curve, where is a regular Lorentzian surface. We define the intrinsic geodesic curvature of at the noncharacteristic point to be if the limit exists.

Proposition 14. Let be a -smooth regular curve, where is a regular Lorentzian surface. (1)If be a spacelike -smooth curve, then and(2)If be a timelike -smooth curve, then andif

Proof. By Equations (61), (62), and (77), we get the following: By Equation (64) and the above equation, we have the following: So, we get the following: Furthermore, When , and , we get the following: where does not depend on . So, When , and , we have the following: We get the following:

Example 15. We take and as in Example 11. Then, By Equation (77), we have the following: Then, So, we have when , then for the large and we have is a spacelike vector. If , by Proposition 14 (1), we have the following: When we take , we have the following: By Equation (77), we have the following: Then, So, we have when , then for the large and we have is a spacelike vector. If , by Proposition 14 (1), we have the following:

4. Curvatures for Lorentzian Surfaces in Lorentzian Approximations

In this section, we will compute the intrinsic Gaussian curvature of Lorentzian surfaces in . We define the second fundamental form of the embedding of into by the following:

We have the following theorem.

Theorem 16. The second fundamental form of the embedding of into is given by the following: where

Proof. Since , we have the following: Using the definition of the connection, identities in Equation (11), and grouping terms, we have the following: Since , we have . Furthermore, we obtain the following: To compute and , using the properties of connection, we compute the following: Next, we compute the inner product of this with . Using the product rule and the identity , we obtain the following: The identities and yield the following: Finally, by using the identity , we obtain the following: Therefore, Since , by similar computation as above, we get the following: Taking the inner product with yields the following: Under some simplifications, one can get the following:

The Riemannian mean curvature of is defined by the following:

Proposition 17. Away from characteristic point, the horizontal mean curvature of is given by the following:

Proof. By we get Equation (139).

Recalling the definition of curvature tensor for a connection is defined by the following:

By Proposition 1 and Equation (141), we have the following lemma.

Lemma 18. The curvature tensor of is given by the following: Let

By the Gauss equation, we have the following:

Proposition 19. Away from characteristic points, we have the following:

Proof. We compute the following: By Theorem 16 and as , we get the following: By Equations (144), (147), and (148), we get the desired equation.

5. The First Gauss-Bonnet Theorem in

In this section, we will prove Gauss-Bonnet theorem in . We consider a spacelike curve , and define the Riemannian length measure by

Lemma 20. Let be a spacelike -smooth. Let Then, When , we have the following: When , we have the following:

Proof. Since we get the following: When , we have the following: Using the Taylor expansion, we can prove the following: From the definition of and , we get the following:

Proposition 21. Let be a Euclidean -smooth surface and and denote the surface measure on with respect to the Lorentzian metric Let Then, If with , then where

Proof. Let We define then, Therefore, Recalling and the Taylor expansion we get Equation (159). By Equation (8), we have the following: where and Let We know that , and so by the dominated convergence theorem, we get the following:

Similar to the proof of Theorem 4.3 in [3], we get a Gauss-Bonnet theorem in as follows:

Theorem 22. Let be a regular Lorentzian surface with finitely many boundary components , given by -smooth regular and closed spacelike curve Let be intrinsic Gaussian curvature of in Proposition 19 and the intrinsic signed geodesic curvature of relative to in Proposition 14. Suppose that the characteristic set satisfies and that is locally summable with respect to the -dimensional Hausdorff measure near the characteristic set . Then,

Proof. Using the discussions in [1, 2], we may assume that all points satisfy the following: Then, by Lemma 20, we obtain the following: By the Gauss-Bonnet theorem, we have the following: So, by Equations (171), (172), (145), (159), and (160) and Lemma 20, we get the following: where . Let go to the infinity, and by using the dominated convergence theorem, we get the desired result.

6. Curvatures for Spacelike Surfaces and the Second Gauss-Bonnet Theorem

The geodesic curvature of spacelike curves on spacelike surface and intrinsic Gaussian curvature of spacelike surfaces in will be investigated in this section. For a regular surface and regular curves , suppose that there is a -smooth function such that

Similar to Section 4, we give We consider the case that is a spacelike surface in . In particular, let , , and Let , when , we have We then define the following:

In particular, . These functions are well defined at every noncharacteristic point. Let then is the unit timelike normal vector to and , is the unit spacelike vector. are the orthonormal basis of . We call a spacelike surface in the Lorentzian group of rigid motions of the Minkowski plane. We define a linear transformation on by ,

For every , we define , where is the projection. Then, is the Levi-Civita connection on with respect to the metric . In particular,

A simple computation shows that

Moreover, if , then

Definition 23. Let be a regular spacelike surface, be a regular -smooth spacelike curve. The geodesic curvature of at is defined as follows:

Definition 24. Let be a regular spacelike surface and be a regular -smooth spacelike curve. The intrinsic geodesic curvature of at is defined to be if the limit exists.

Proposition 25. Let be a regular spacelike surface and be a regular -smooth spacelike curve. Then,

Proof. By Equation (25) and , we get the following: By Equation (181), we have the following: Similarly, we have that when , By Equations (181) and (189), we have the following: where does not depend on . By Equation (183), we have the following: When , and , we have the following: By Equations (194)–(196) and Equation (182), we get the following: When and , we have the following: if , and , so we get Equation (188).

Proposition 26. Let be a regular spacelike surface. Let be a spacelike -smooth regular curve. Then, if , and

Proof. By Equations (176) and (188), we get the following: By Equation (180) and the above equation, we have the following: So, we get the following: Furthermore, When , and , we get the following: where does not depend on . So, When , and , we have the following: as We get the following:

In the following, we compute the intrinsic Gaussian curvature of spacelike surfaces in . Similar to Theorem 4.3 in [16], we have the following:

Theorem 27. The second fundamental form of the embedding of into is given by the following: where

Proof. Since , we have the following: Using the definition of the connection, the identities in Equation (11), and grouping terms, we have the following: Since , we have . We have the following: Similarly, we have the following:

Similar to Proposition 17, we get the expression of the horizontal mean curvature of the spacelike surface.

Proposition 28. Away from characteristic point, the horizontal mean curvature of is given by the following:

Proposition 29. Away from characteristic points, we have the following:

Proof. By Equation (141) and Lemma 18, we have the following: Similar to Equation (148), we obtain the following: By Equations (215) and (216) and , we get the desired equation.
Similar to Lemma 20 and Proposition 21, for spacelike curve and spacelike surface, we obtain the following: Combining Equations (214) and (217) and Proposition 26, similar to the proof of Theorem 22, we have the second Gauss-Bonnet theorem.

Theorem 30. Let be a regular spacelike surface with finitely many boundary components , given by -smooth closed and regular spacelike curves Let be intrinsic Gaussian curvature of in Proposition 29 and the intrinsic signed geodesic curvature of relative to in Proposition 26. Suppose that the characteristic set satisfies and that is locally summable with respect to the -dimensional Hausdorff measure near the characteristic set . Then,

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

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

Acknowledgments

This research is supported by the Natural Science Foundation of Heilongjiang Province of China (Grant No. LH2021A020).

References

Z. Balogh, J. Tyson, and E. Vecchi, “Intrinsic curvature of curves and surfaces and a Gauss-Bonnet theorem in the Heisenberg group,” Mathematische Zeitschrift, vol. 287, no. 1-2, pp. 1–38, 2017.
View at: Publisher Site | Google Scholar
Z. Balogh, J. Tyson, and E. Vecchi, “Correction to: Intrinsic curvature of curves and surfaces and a Gauss-Bonnet theorem in the Heisenberg group,” Mathematische Zeitschrift, vol. 296, no. 1-2, pp. 875-876, 2020.
View at: Publisher Site | Google Scholar
Y. Wang and S. Wei, “Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane,” Science China Mathematics, vol. 64, pp. 1843–1860, 2021.
View at: Publisher Site | Google Scholar
S. Wei and Y. Wang, “Gauss-Bonnet theorems in the Lorentzian Heisenberg group and the Lorentzian group of rigid motions of the Minkowski plane,” Symmetry, vol. 13, p. 173, 2020.
View at: Publisher Site | Google Scholar
Y. Wang and S. Wei, “Gauss-Bonnet theorems in the BCV spaces and the twisted Heisenberg group,” Results in Mathematics, vol. 75, 2020.
View at: Publisher Site | Google Scholar
T. Wu, S. Wei, and Y. Wang, “Gauss-Bonnet theorems and the Lorentzian Heisenberg group,” Turkish Journal of Mathematics, vol. 45, no. 2, pp. 718–741, 2021.
View at: Publisher Site | Google Scholar
H. Liu, J. Miao, W. Li, and J. Guan, “The Sub-Riemannian limit of curvatures for curves and surfaces and a Gauss–Bonnet theorem in the rototranslation group,” Journal of Mathematics, vol. 2021, Article ID 9981442, 22 pages, 2021.
View at: Publisher Site | Google Scholar
W. Li and H. Liu, “Gauss-Bonnet theorem in the universal covering group of Euclidean motion group with the general left-invariant metric,” Journal of Nonlinear Mathematical Physics, vol. 29, no. 1, pp. 1–32, 2022.
View at: Publisher Site | Google Scholar
H. Liu and J. Miao, “Gauss-Bonnet theorem in Lorentzian Sasakian space forms,” AIMS Mathematics, vol. 6, no. 8, pp. 8772–8791, 2021.
View at: Publisher Site | Google Scholar
J. Guan and H. Liu, “The sub-Riemannian limit of curvatures for curves and surfaces and a Gauss-Bonnet theorem in the group of rigid motions of Minkowski plane with general left-invariant metric,” Journal of Function Spaces, vol. 2021, Article ID 1431082, 14 pages, 2021.
View at: Publisher Site | Google Scholar
K. Nomizu, “Left-invariant Lorentz metrics on Lie groups,” Osaka Journal of Mathematics, vol. 16, pp. 143–150, 1979.
View at: Google Scholar
K. Onda, “Lorentz Ricci solitons on 3-dimensional Lie groups,” Geometriae Dedicata, vol. 147, no. 1, pp. 313–322, 2010.
View at: Publisher Site | Google Scholar
V. Patrangenaru, “Classifying 3- and 4-dimensional homogeneous Riemannian manifolds by Cartan triples,” Pacific Journal of Mathematics, vol. 173, no. 2, pp. 511–532, 1996.
View at: Publisher Site | Google Scholar
J. I. Inoguchi and J. Van der Veken, “Parallel surfaces in the motion groups and ,” Bulletin of the Belgian Mathematical Society, vol. 14, no. 2, pp. 321–332, 2007.
View at: Publisher Site | Google Scholar
J. I. Inoguchi and J. Van der Veken, “A complete classification of parallel surfaces in three-dimensional homogeneous spaces,” Geometriae Dedicata, vol. 131, no. 1, pp. 159–172, 2008.
View at: Publisher Site | Google Scholar
L. Capogna, D. Danielli, S. Pauls, and D. Danielli, “An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem,” Progress in Mathematics, Birkhäuser Verlag, Basel, vol. 259, 2007.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Haiming Liu and Jianyun Guan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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