Lorentzian approximations for a Lorentzian $ \alpha $-Sasakian manifold and Gauss-Bonnet theorems

Haiming Liu; Xiawei Chen; Jianyun Guan; Peifu Zu; Haiming Liu; Xiawei Chen; Jianyun Guan; Peifu Zu

doi:10.3934/math.2023024

AIMS Mathematics

2023, Volume 8, Issue 1: 501-528. doi: 10.3934/math.2023024

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Research article

Lorentzian approximations for a Lorentzian $ \alpha $-Sasakian manifold and Gauss-Bonnet theorems

School of Mathematics, Mudanjiang Normal University, Mudanjiang 157011, China

Received: 29 May 2022 Revised: 16 September 2022 Accepted: 25 September 2022 Published: 09 October 2022
MSC : 53C40, 53C42

In this paper, we define the Lorentzian approximations of a $ 3 $-dimensional Lorentzian $ \alpha $-Sasakian manifold. Moreover, we define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on Lorentzian surfaces and spacelike surfaces and the intrinsic Gaussian curvature of Lorentzian surfaces and spacelike surfaces away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove Gauss-Bonnet theorems for the Lorentzian surfaces and spacelike surfaces in the Lorentzian $ \alpha $-Sasakian manifold.
- Lorentzian $ \alpha $-Sasakian manifold,
- Gauss-Bonnet theorem,
- Lorentzian approximations,
- sub-Lorentzian geometry,
- intrinsic curvature
Citation: Haiming Liu, Xiawei Chen, Jianyun Guan, Peifu Zu. Lorentzian approximations for a Lorentzian $ \alpha $-Sasakian manifold and Gauss-Bonnet theorems[J]. AIMS Mathematics, 2023, 8(1): 501-528. doi: 10.3934/math.2023024

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Abstract

In this paper, we define the Lorentzian approximations of a $ 3 $-dimensional Lorentzian $ \alpha $-Sasakian manifold. Moreover, we define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on Lorentzian surfaces and spacelike surfaces and the intrinsic Gaussian curvature of Lorentzian surfaces and spacelike surfaces away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove Gauss-Bonnet theorems for the Lorentzian surfaces and spacelike surfaces in the Lorentzian $ \alpha $-Sasakian manifold.

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