Generalized Bishop frames of regular time-like curves in 4-dimensional Lorentz space $\mathbb{L}^{4}$

TL;DR

This paper extends the theory of generalized Bishop frames to regular time-like curves in four-dimensional Lorentz space $\mathbb{L}^{4}$ by introducing a classification of frames whose coefficient matrices have at most three nonzero entries above the diagonal, yielding four types (B,C,D,F) up to permutation. It proves a Lorentzian Bishop-type existence result (type B) for all $C^{2}$ time-like curves and establishes a hierarchical relation among frame types (F ⇒ D ⇒ C), with type B corresponding to the classic Bishop frame and type F corresponding to Frenet-like behavior. The authors show the hierarchy is proper via explicit counterexamples and develop a transformation framework $G' = X_{1}G - G X_{0}$ to relate frames, enabling a systematic reduction of frame-existence problems to ODEs. These results yield a Lorentzian analogue of the Euclidean hierarchy and provide a new curve-classification tool for time-like curves in $\mathbb{L}^{4}$. The work has potential implications for the geometric analysis of Lorentzian curves and their applications in theoretical physics and computer graphics on Lorentzian manifolds.

Abstract

We introduced generalized Bishop frames on curves in 4-dimensional Euclidean space $\mathbb{E}^{4}$, which are orthonormal frames such that the derivatives of the vectors of the frames along the curve can be expressed, via a certain matrix, as a linear combination of the vectors of the frame. In relation to that, we study generalized Bishop frames of regular time-like curves. In a previous work, we showed that there is a hierarchy among different types of generalized Bishop frames for regular curves in the Euclidean space. Building upon this study, we further investigate it in the 4-dimensional Lorentz space $\mathbb{L}^4$. There are four types of generalized Bishop frames of regular time-like curves in $\mathbb{L}^{4}$ up to the change of the order of vectors fixing the first one which is the tangent vector. Unlike other types of curves, such as light-like and space-like ones, the time-like curve can be investigated in a manner analogous to the Euclidean case. We find that a hierarchy of frames exists, similar to that in the Euclidean setting. Based on this hierarchy, we propose a new classification of curves.

Theorems & Definitions (26)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4: NN
• Corollary 1.5
• Theorem 1.6
• Definition 2.1: bo
• Lemma 2.2
• Lemma 2.3
• proof