Generalized Timelike Mannheim Curves in Minkowski Space-Time 𝐸^𝟒𝟏

Abstract

We give the definition of generalized timelike Mannheim curve in Minkowski space-time 𝐸^𝟒𝟏. The necessary and sufficient conditions for the generalized timelike Mannheim curve are obtained. We show some characterizations of generalized Mannheim curve.

1. Introduction

The geometry of curves has long captivated the interests of mathematicians, from the ancient Greeks through to the era of Isaac Newton (1643–1727) and the invention of the calculus. It is a branch of geometry that deals with smooth curves in the plane and in the space by methods of differential and integral calculus. The theory of curves is the simpler and narrower in scope because a regular curve in a Euclidean space has no intrinsic geometry. One of the most important tools used to analyze curve is the Frenet frame, a moving frame that provides a coordinate system at each point of curve that is "best adopted" to the curve near that point. Every person of classical differential geometry meets early in his course the subject of Bertrand curves, discovered in 1850 by J. Bertrand. A Bertrand curve is a curve such that its principal normals are the principal normals of a second curve. There are many works related with Bertrand curves in the Euclidean space and Minkowski space, [1–3].

Another kind of associated curve is called Mannheim curve and Mannheim partner curve. The notion of Mannheim curves was discovered by A. Mannheim in 1878. These curves in Euclidean 3-space are characterized in terms of the curvature and torsion as follows: a space curve is a Mannheim curve if and only if its curvature 𝑘₁ and torsion 𝑘₂ satisfy the relation𝑘₁=𝛽𝑘²¹+𝑘²²(1.1) for some constant 𝛽. The articles concerning Mannheim curves are rather few. In [4], a remarkable class of Mannheim curves is studied. General Mannheim curves in the Euclidean 3-space are obtained in [5–7]. Recently, Mannheim curves are generalized and some characterizations and examples of generalized Mannheim curves are given in Euclidean 4-space 𝐸⁴ by [8].

In this paper, we study the generalized timelike Mannheim partner curves in 4-dimensional Minkowski space-time. We will give the necessary and sufficient conditions for the generalized timelike Mannheim partner curves.

2. Preliminaries

To meet the requirements in the next sections, the basic elements of the theory of curves in Minkowski space-time 𝐸⁴¹ are briefly presented in this section. A more complete elementary treatment can be found in [9].

Minkowski space-time 𝐸⁴¹ is a usual vector space provided with the standard flat metric given by ⟨,⟩=−𝑑𝑥²¹+𝑑𝑥²²+𝑑𝑥²³+𝑑𝑥²⁴,(2.1) where (𝑥₁,𝑥₂,𝑥₃,𝑥₄) is a rectangular coordinate system in 𝐸⁴¹.

Since ⟨,⟩ is an indefinite metric, recall that a 𝐯∈𝐸⁴¹ can have one of the three causal characters; it can be spacelike if ⟨𝐯,𝐯⟩>0 or 𝐯=0, timelike if ⟨𝐯,𝐯⟩<0, and null(ligthlike) if ⟨𝐯,𝐯⟩=0 and 𝐯≠0. Similarly, an arbitrary curve 𝐜=𝐜(𝑡) in 𝐸⁴¹ can locally be spacelike, timelike, or null (lightlike) if all of its velocity vectors 𝐜^(𝑡) are, respectively, spacelike, timelike, or null. The norm of 𝐯∈𝐸⁴¹ is given by ‖𝐯‖=√|⟨𝐯,𝐯⟩|. If ‖𝐜^(𝑡)‖=√|⟨𝐜^(𝑡),𝐜^(𝑡)⟩|≠0 for all 𝑡∈𝐼, then 𝐶 is a regular curve in 𝐸⁴¹. A timelike (spacelike) regular curve 𝐶 is parameterized by arc-length parameter 𝑡 which is given by 𝐜∶𝐼→𝐸⁴¹, then the tangent vector 𝐜^(𝑡) along 𝐶 has unit length, that is, ⟨𝐜^(𝑡),𝐜^(𝑡)⟩=−1, (⟨𝐜^(𝑡),𝐜^(𝑡)⟩=1) for all 𝑡∈𝐼.

Hereafter, curves considered are timelike and regular 𝐶^∞ curves in 𝐸⁴¹. Let 𝐓(𝑡)=𝐜^(𝑡) for all 𝑡∈𝐼; then the vector field 𝐓(𝑡) is timelike and it is called timelike unit tangent vector field on 𝐶.

The timelike curve 𝐶 is called special timelike Frenet curve if there exist three smooth functions 𝑘₁, 𝑘₂, 𝑘₃ on 𝐶 and smooth nonnull frame field {𝐓,𝐍,𝐁₁,𝐁₂} along the curve 𝐶. Also, the functions 𝑘₁, 𝑘₂, and 𝑘₃ are called the first, the second, and the third curvature function on 𝐶, respectively. For the 𝐶^∞ special timelike Frenet curve 𝐶, the following Frenet formula is⎡⎢⎢⎢⎢⎣𝐓^𝐍^𝐁^1𝐁^2⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎣𝑘0𝑘¹00₁0−0𝑘₂𝑘₂00𝑘₃00−𝑘₃0⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣𝐁𝐁𝐍𝐓¹₂⎤⎥⎥⎥⎦,(2.2)see [9].

Here, due to characters of Frenet vectors of the timelike curve, 𝐓, 𝐍, 𝐁₁, and 𝐁₂ are mutually orthogonal vector fields satisfying equations ⟨𝐓,𝐓⟩=−1,⟨𝐍,𝐍⟩=⟨𝐁₁,𝐁₁⟩=⟨𝐁₂,𝐁₂⟩=1.(2.3) For 𝑡∈𝐼, the nonnull frame field {𝐓,𝐍,𝐁₁,𝐁₂} and curvature functions 𝑘₁, 𝑘₂, and 𝑘₃ are determined as follows:21ststep𝐓(𝑡)=𝐜^(𝑡),ndstep𝐍𝑘₁((𝑡𝑡))==‖‖𝐓1^(𝑡)‖‖>0,𝑘₁(𝑡)𝐓^3rd(𝑡),step𝑘𝐁₂(𝑡)=‖‖𝐍^(𝑡)−𝑘₁(𝑡)𝐓(𝑡)‖‖>0,₁(𝑡)=1𝑘₂(𝑡)𝐍^4(𝑡)−𝑘₁(𝑡)𝐓(𝑡),thstep𝐁₂(𝑡)=𝛿1‖‖𝐁^1(𝑡)+𝑘₂(𝑡)𝐍(𝑡)‖‖𝐁^1(𝑡)+𝑘₂(𝑡)𝐍(𝑡),(2.4) where 𝛿 is determined by the fact that orthonormal frame field {𝐓(𝑡),𝐍(𝑡),𝐁₁(𝑡),𝐁₂(𝑡)} is of positive orientation. The function 𝑘₃ is determined by𝑘₃(𝑡)=𝐁^1(𝑡),𝐁₂(𝑡)≠0.(2.5) So the function 𝑘₃ never vanishes.

In order to make sure that the curve 𝐶 is a special timelike Frenet curve, above steps must be checked, from 1st step to 4th step, for 𝑡∈𝐼.

Let {𝐓,𝐍,𝐁₁,𝐁₂} be the moving Frenet frame along a unit speed timelike curve 𝐶 in 𝐸⁴¹, consisting of the tangent, the principal normal, the first binormal, and the second binormal vector field, respectively. Since 𝐶 is a timelike curve, its Frenet frame contains only nonnull vector fields.

3. Generalized Timelike Mannheim Curves in 𝐸⁴¹

Mannheim curves are generalized by Matsuda and Yorozu in [8]. In this paper, we have investigated the generalization of timelike Mannheim curves in Minkowski space 𝐸⁴¹.

Definition 3.1. A special timelike curve 𝐶 in 𝐸⁴¹ is a generalized timelike Mannheim curve if there exists a special timelike Frenet curve 𝐶^∗ in 𝐸⁴¹ such that the first normal line at each point of 𝐶 is included in the plane generated by the second normal line and the third normal line of 𝐶^∗ at the corresponding point under 𝜙. Here, 𝜙 is a bijection from 𝐶 to 𝐶^∗. The curve 𝐶^∗ is called the generalized timelike Mannheim mate curve of 𝐶.

By the definition, a generalized Mannheim mate curve 𝐶^∗ is given by the map 𝐜^∗∶𝐼^∗→𝐸⁴¹ such that𝐜^∗(𝑡)=𝐜(𝑡)+𝛽(𝑡)𝐍(𝑡),𝑡∈𝐼.(3.1) Here 𝛽 is a smooth function on 𝐼. Generally, the parameter 𝑡 is not an arc-length of 𝐶^∗. Let 𝑡^∗ be the arc-length of 𝐶^∗ defined by 𝑡^∗=^𝑡0‖‖‖𝑑𝐜^∗(𝑡)𝑑𝑡‖‖‖𝑑𝑡.(3.2) If a smooth function 𝑓∶𝐼→𝐼^∗ is given by 𝑓(𝑡)=𝑡^∗, then for all 𝑡∈𝐼, we have 𝑓^(𝑡)=𝑑𝑡^∗𝑑𝑡=‖‖‖𝑑𝐜^∗(𝑡)𝑑𝑡‖‖‖=|||−1+𝛽(𝑡)𝑘₁(𝑡)²+𝛽^(𝑡)²+𝛽(𝑡)𝑘₂(𝑡)²|||.(3.3) The representation of timelike curve 𝐶^∗ with arc-length parameter 𝑡^∗ is 𝐜^∗∶𝐼𝑡^∗_∗⟶𝐸⁴¹,⟶𝐜^∗𝑡^∗.(3.4) For a bijection 𝜙∶𝐶→𝐶^∗ defined by 𝜙(𝐜(𝑡))=𝐜^∗(𝑓(𝑡)), the reparameterization of 𝐶^∗ is𝐜^∗(𝑓(𝑡))=𝐜(𝑡)+𝛽(𝑡)𝐍(𝑡),(3.5) where 𝛽 is a smooth function on 𝐼. Thus, we have 𝑑𝐜^∗(𝑓(𝑡))𝑑𝑡=𝑑𝐜^∗𝑡^∗𝑑𝑡||||_{𝑡∗=𝑓(𝑡)}𝑓^(𝑡)=𝑓^(𝑡)𝐓^∗(𝑓(𝑡)),𝑡∈𝐼.(3.6)

Theorem 3.2. If a special timelike Frenet curve 𝐶 in 𝐸⁴¹ is a generalized timelike Mannheim curve, then the following relation between the first curvature function 𝑘₁ and the second curvature function 𝑘₂ holds: 𝑘₁(𝑡)=−𝛽𝑘²¹(𝑡)−𝑘²²(𝑡),𝑡∈𝐼,(3.7) where 𝛽 is a constant number.

Proof. Let 𝐶 be a generalized timelike Mannheim curve and 𝐶^∗ the generalized timelike Mannheim mate curve of 𝐶, as the following diagram: (3.8) A smooth function 𝑓 is defined by 𝑓(𝑡)=∫‖𝑑𝐜^∗(𝑡)/𝑑𝑡‖𝑑𝑡=𝑡^∗ and 𝑡^∗ is the arc-length parameter of 𝐶^∗. Also, 𝜙 is a bijection defined by 𝜙(𝐜(𝑡))=𝐜^∗(𝑓(𝑡)). Thus, the timelike curve 𝐶^∗ is reparametrized as follows 𝐜^∗(𝑓(𝑡))=𝐜(𝑡)+𝛽(𝑡)𝐍(𝑡),(3.9) where 𝛽∶𝐼⊂ℝ→ℝ is a smooth function. By differentiating both sides of (3.9) with respect to 𝑡, we have 𝑓^(𝑡)𝐓^∗(𝑓(𝑡))=1+𝛽(𝑡)𝑘₁(𝑡)𝐓+𝛽^(𝑡)𝐍(𝑡)+𝛽(𝑡)𝑘₂(𝑡)𝐁₁(𝑡).(3.10) On the other hand, since the first normal line at each point of 𝐶 is lying in the plane generated by the second normal line and the third normal line of 𝐶^∗ at the corresponding points under bijection 𝜙, the vector field 𝐍(𝑡) is given by 𝐍(𝑡)=𝑔(𝑡)𝐁_1∗(𝑓(𝑡))+ℎ(𝑡)𝐁_2∗(𝑓(𝑡)),(3.11) where 𝑔 and ℎ are some smooth functions on 𝐼⊂ℝ. If we take into consideration ⟨𝐓^∗(𝑓(𝑡)),𝑔(𝑡)𝐁_1∗(𝑓(𝑡))+ℎ(𝑡)𝐁_2∗(𝑓(𝑡))⟩=0(3.12) and (3.10), then we have 𝛽^(𝑡)=0. So we rewrite (3.10) as 𝑓^(𝑡)𝐓^∗(𝑓(𝑡))=1+𝛽𝑘₁(𝑡)𝐓(𝑡)+𝛽𝑘₂(𝑡)𝐁₁(𝑡),(3.13) that is, 𝐓^∗(𝑓(𝑡))=1+𝛽𝑘¹(𝑡)𝑓^(𝑡)𝐓(𝑡)+𝛽𝑘²(𝑡)𝑓^(𝑡)𝐁¹(𝑡),(3.14) where 𝑓^(𝑡)=|||−1+𝛽𝑘₁(𝑡)²+𝛽𝑘₂(𝑡)²|||.(3.15) By taking the differentiations both sides of (3.13) with respect to 𝑡∈𝐼, we get 𝑓^(𝑡)𝑘^∗1(𝑓(𝑡))𝐍^∗(𝑓(𝑡))=1+𝛽𝑘¹(𝑡)𝑓^(𝑡)^+1+𝛽𝑘₁𝐓(𝑡)(𝑡)𝑘₁(𝑡)−𝛽𝑘₂(𝑡)²+𝛽𝑘₂𝑓^(𝑡)𝐍(𝑡)(𝑡)𝑓^(𝑡)^𝐁₁(𝑡)+𝛽𝑘²(𝑡)𝑘₃(𝑡)𝑓^(𝑡)𝐁²(𝑡).(3.16) Since ⟨𝐍^∗(𝑓(𝑡)),𝑔(𝑡)𝐁_1∗(𝑓(𝑡))+ℎ(𝑡)𝐁_2∗(𝑓(𝑡))⟩=0,(3.17) the coefficient of 𝐍(𝑡) in (3.16) vanishes, that is, 1+𝛽𝑘₁(𝑡)𝑘₁(𝑡)−𝛽𝑘₂(𝑡)²=0.(3.18) Thus, this completes the proof.

Theorem 3.3. In 𝐸⁴¹, let 𝐶 be a special timelike Frenet curve such that its nonconstant first and second curvature functions satisfy the equality 𝑘₁(𝑠)=−𝛽(𝑘²¹(𝑡)−𝑘²²(𝑡)) for all 𝑡∈𝐼⊂ℝ. If the timelike curve 𝐶^∗ given by 𝐜^∗(𝑡)=𝐜(𝑡)+𝛽𝐍(𝑡)(3.19) is a special timelike Frenet curve, then 𝐶^∗ is a generalized timelike Mannheim mate curve of 𝐶.

Proof. The arc-length parameter of 𝐶^∗ is given by 𝑡^∗=^𝑡0‖‖‖𝑑𝐜^∗(𝑡)𝑑𝑡‖‖‖𝑑𝑡,𝑡∈𝐼.(3.20) Under the assumption of 𝑘₁(𝑡)=−𝛽𝑘²¹(𝑡)−𝑘²²(𝑡),(3.21) we obtain 𝑓^(𝑡)=√|1+𝛽𝑘₁(𝑡)|, 𝑡∈𝐼.
Differentiating the equation 𝐜^∗(𝑓(𝑡))=𝐜(𝑡)+𝛽𝐍(𝑡) with respect to 𝑡, we reach𝑓^(𝑡)𝐓^∗(𝑓(𝑡))=1+𝛽𝑘₁(𝑡)𝐓(𝑡)+𝛽𝑘₂(𝑡)𝐁₁(𝑡).(3.22) Thus, it is seen that 𝐓^∗(𝑓(𝑡))=⎛⎜⎜⎜⎝1+𝛽𝑘¹(𝑡)||1+𝛽𝑘₁(𝑡)||𝐓(𝑡)+𝛽𝑘²(𝑡)||1+𝛽𝑘₁(𝑡)||𝐁¹(𝑡)⎞⎟⎟⎟⎠,𝑡∈𝐼.(3.23) The differentiation of the last equation with respect to 𝑡 is 𝑓^(𝑡)𝑘^∗1(𝑓(𝑡))𝐍^∗(𝑓(𝑡))=+⎛⎜⎜⎜⎝1||1++𝛽𝛽𝑘𝑘₁₁(𝑡)||^𝐓(𝑡)(𝑡)𝑘₁(𝑡)−𝛽𝑘²²(𝑡)+⎛⎜⎜⎜⎝𝛽𝑘₂||1+𝛽𝑘₁(𝑡)||⎞⎟⎟⎟⎠𝐍(𝑡)(𝑡)||1+𝛽𝑘₁(𝑡)||⎞⎟⎟⎟⎠^𝐁₁(𝑡)+⎛⎜⎜⎜⎝𝛽𝑘²(𝑡)𝑘₃(𝑡)||1+𝛽𝑘₁(𝑡)||⎞⎟⎟⎟⎠𝐁²(𝑡).(3.24) From our assumption, we have 𝑘₁(𝑡)+𝛽𝑘²¹(𝑡)−𝛽𝑘²²(𝑡)||1+𝛽𝑘₁(𝑡)||=0.(3.25) Thus, the coefficient of 𝐍(𝑡) in (3.24) is zero. It is seen from (3.23) that 𝐓^∗(𝑓(𝑡)) is a linear combination of 𝐓(𝑡) and 𝐁₁(𝑡). Additionally, from (3.24), 𝐍^∗(𝑓(𝑡)) is given by linear combination of 𝐓(𝑡), 𝐁₁(𝑡), and 𝐁₂(𝑡). On the other hand, 𝐶^∗ is a special timelike Frenet curve that the vector 𝐍(𝑡) is given by linear combination of 𝐁^∗1(𝑓(𝑡)) and 𝐁^∗2(𝑓(𝑡)).
Therefore, the first normal line of 𝐶 lies in the plane generated by the second normal line and third normal line of 𝐶^∗ at the corresponding points under a bijection 𝜙 defined by 𝜙(𝐜(𝑡))=𝐜^∗(𝑓(𝑡)).
This completes the proof.

Remark 3.4. In 4-dimensional Minkowski space 𝐸⁴¹, a special timelike Frenet curve 𝐶 with curvature functions 𝑘₁ and 𝑘₂ satisfying 𝑘₁(𝑡)=−𝛽(𝑘²¹(𝑡)−𝑘²²(𝑡)), it is not clear that a smooth timelike curve 𝐶^∗ given by (3.1) is a special Frenet curve. Thus, it is unknown whether the reverse of Theorem 3.2 is true or not.

Theorem 3.5. Let 𝐶 be a special timelike curve in 𝐸⁴¹ with nonzero third curvature function 𝑘₃. There exists a timelike special Frenet curve 𝐶^∗ in 𝐸⁴¹ such that the first normal line of 𝐶 is linearly dependent with the third normal line of 𝐶^∗ at the corresponding points 𝐜(𝑡) and 𝐜^∗(𝑡), respectively, under a bijection 𝜙∶𝐶→𝐶^∗, if and only if the curvatures 𝑘₁ and 𝑘₂ of 𝐶 are constant functions.

Proof. Let 𝐶 be a timelike Frenet curve in 𝐸⁴¹ with the Frenet frame field {𝐓,𝐍,𝐁₁,𝐁₂} and curvature functions 𝑘₁, 𝑘₂, and 𝑘₃. Also, we assume that 𝐶^∗ is a timelike special Frenet curve in 𝐸⁴¹ with the Frenet frame field {𝐓^∗,𝐍^∗,𝐁_1∗,𝐁_2∗} and curvature functions 𝑘^∗1, 𝑘^∗2, and 𝑘^∗3. Let the first normal line of 𝐶 be linearly dependent with the third normal line of 𝐶^∗ at the corresponding points 𝐶 and 𝐶^∗, respectively. Then the parameterization of 𝐶^∗ is 𝐜^∗(𝑓(𝑡))=𝐜(𝑡)+𝛽(𝑡)𝐍(𝑡),𝑡∈𝐼.(3.26) If the arc-length parameter of 𝐶^∗ is given 𝑡^∗, then 𝑡^∗=^𝑡0|||−1+𝛽(𝑡)𝑘₁(𝑡)²+𝛽^(𝑡)²+𝛽(𝑡)𝑘₂(𝑡)²|||𝑑𝑡,(3.27)𝑓∶𝐼𝑡⟶⟶𝑓𝐼^∗(,𝑡)=𝑡_∗.(3.28) Moreover, 𝜙∶𝐶→𝐶^∗ is a bijection given by 𝜙(𝐜(𝑡))=𝐜^∗(𝑓(𝑡)).
Differentiating (3.26) with respect to 𝑡 and using the Frenet formulas, we get𝑓^(𝑡)𝐓^∗(𝑓(𝑡))=1+𝛽(𝑡)𝑘₁(𝑡)𝐓(𝑡)+𝛽^(𝑡)𝐍(𝑡)+𝛽(𝑡)𝑘₂(𝑡)𝐁₁(𝑡).(3.29) Since 𝐁_2∗(𝑓(𝑡))=∓𝐍(𝑡), then 𝑓^(𝑡)𝐓^∗(𝑓(𝑡)),𝐁_2∗(𝑓(𝑡))=⟨1+𝛽(𝑡)𝑘₁(𝑡)𝐓(𝑡)+𝛽^(𝑡)𝐍(𝑡)+𝛽(𝑡)𝑘₂(𝑡)𝐁₁(𝑡),∓𝐍(𝑡)⟩(3.30) that is, 0=∓𝛽^(𝑡).(3.31) From the last equation, it is easily seen that 𝛽 is a constant number. Hereafter, we can denote 𝛽(𝑡)=𝛽, for all 𝑡∈𝐼.
From (3.27), we have𝑓^(𝑡)=|||−1+𝛽𝑘₁(𝑡)²+𝛽𝑘₂(𝑡)²|||>0.(3.32) Thus, we rewrite (3.29) as follows: 𝐓^∗(𝑓(𝑡))=1+𝛽𝑘¹(𝑡)𝑓^(𝑡)𝐓(𝑡)+𝛽𝑘²(𝑡)𝑓^(𝑡)𝐁¹(𝑡).(3.33) The differentiation of the last equation with respect to 𝑡 is 𝑓^(𝑡)𝑘^∗1(𝑓(𝑡))𝐍^∗(𝑓(𝑡))=1+𝛽𝑘¹(𝑡)𝑓^(𝑡)^+1+𝛽𝑘₁𝐓(𝑡)(𝑡)𝑘₁(𝑡)−𝛽𝑘²²(𝑡)+𝛽𝑘₂𝑓^(𝑡)𝐍(𝑡)(𝑡)𝑓^(𝑡)^𝐁₁(𝑡)+𝛽𝑘²(𝑡)𝑘₃(𝑡)𝑓^(𝑡)𝐁²(𝑡).(3.34) Since ⟨𝑓^(𝑡)𝑘^∗1(𝑓(𝑡))𝐍^∗(𝑓(𝑡)),𝐁_2∗(𝑓(𝑡))⟩=0 and 𝐁_2∗(𝑓(𝑡))=∓𝐍(𝑡) for all 𝑡∈𝐼, 𝑘₁(𝑡)+𝛽𝑘²¹(𝑡)−𝛽𝑘²²(𝑡)=0(3.35) is satisfied. Then 𝛽=−𝑘¹(𝑡)𝑘²¹(𝑡)−𝑘²²(𝑡)(3.36) is a nonzero constant number. Thus, from (3.34), we reach 𝐍^∗(𝑓(𝑡))=1𝑓^(𝑡)𝐾(𝑡)1+𝛽𝑘¹(𝑡)𝑓^(𝑡)^𝐓(𝑡)+1𝑓^(𝑡)𝐾(𝑡)𝛽𝑘²(𝑡)𝑓^(𝑡)^+1𝐁₁(𝑡)𝑓^(𝑡)𝐾(𝑡)𝛽𝑘²(𝑡)𝑘₃(𝑡)𝑓^(𝑡)𝐁²(𝑡),(3.37) where 𝐾(𝑡)=𝑘^∗1(𝑓(𝑡)) for all 𝑡∈𝐼. Differentiating the last equation with respect to 𝑡, then we have 𝑓^(𝑡)=𝑘^∗1(𝑓(𝑡1))𝐓^∗(𝑓(𝑡))+𝑘^∗2(𝑓(𝑡))𝐁^∗1(𝑓(𝑡))𝑓^(𝑡)𝐾(𝑡)1+𝛽𝑘¹(𝑡)𝑓^(𝑡)^^+𝑘𝐓(𝑡)₁(𝑡)𝑓^(𝑡)𝐾(𝑡)1+𝛽𝑘¹(𝑡)𝑓^(𝑡)^−𝑘²(𝑡)𝑓^(𝑡)𝐾(𝑡)𝛽𝑘²(𝑡)+1𝑓^(𝑡)^𝐍(𝑡)𝑓^(𝑡)𝐾(𝑡)𝛽𝑘²(𝑡)𝑓^(𝑡)^^−𝑘³(𝑡)𝑓^(𝑡)𝐾(𝑡)𝛽𝑘²(𝑡)𝑘₃(𝑡)𝑓^(𝑡)𝐁¹+1(𝑡)𝑓^(𝑡)𝐾(𝑡)𝛽𝑘²(𝑡)𝑘₃(𝑡)𝑓^(𝑡)^+𝑘³(𝑡)𝑓^(𝑡)𝐾(𝑡)𝛽𝑘²(𝑡)𝑓^(𝑡)^𝐁²(𝑡)(3.38) for all 𝑡∈𝐼. Considering 𝑓^(𝑡)𝑘^∗1(𝑓(𝑡))𝐓^∗(𝑓(𝐁𝑡))+𝑘^∗2(𝑓(𝑡))𝐁^∗1(𝑓(𝑡)),𝐁^∗2(𝑓(𝑡))=0,_∗2(𝑓(𝑡))=∓𝐍(𝑡),(3.39) then we get 𝑘₁(𝑡)1+𝛽𝑘¹(𝑡)𝑓^(𝑡)^−𝑘₂(𝑡)𝛽𝑘²(𝑡)𝑓^(𝑡)^=0.(3.40) Arranging the last equation, we find 𝛽𝑘₁(𝑡)𝑘^1(𝑡)−𝑘₂(𝑡)𝑘^2(𝑡)𝑓^(𝑡)−𝑘₁(𝑡)+𝛽𝑘²¹(𝑡)−𝛽𝑘²²(𝑡)𝑓^(𝑡)=0.(3.41) Moreover, the differentiation of (3.36) with respect to 𝑡 is 𝑘^1(𝑡)+2𝛽𝑘₁(𝑡)𝑘^1(𝑡)−𝑘₂(𝑡)𝑘^2(𝑡)=0.(3.42) From the above equation, it is seen that −𝑘^1(𝑡)2=𝛽𝑘¹(𝑡)𝑘^1(𝑡)−𝑘₂(𝑡)𝑘^2(𝑡).(3.43) Substituting (3.36) and (3.43) into (3.41), we obtain −𝑘^1(𝑡)2=0.(3.44) This means that the first curvature function is constant (that is, positive constant). Additionally, from (3.43) it is seen that the second curvature function 𝑘₂ is positive constant, too.
Conversely, suppose that 𝐶 is a timelike Frenet curve in 𝐸⁴¹ with the Frenet frame field {𝐓,𝐍,𝐁₁,𝐁₂} and curvature functions 𝑘₁, 𝑘₂, and 𝑘₃. The first curvature function 𝑘₁ and the second curvature function 𝑘₂ of 𝐶 are of positive constant. Thus, 𝑘₁/(𝑘²²−𝑘²¹) is a positive constant number, say 𝛽.
The representation of timelike curve 𝐶^∗ with arc-length parameter 𝑡 is𝐜^∗∶𝐼𝑡⟶⟶𝐜𝐸⁴¹_∗,(𝑡)=𝐜(𝑡)+𝛽(𝑡)𝐍(𝑡).(3.45) Let 𝑡^∗ denote the arc-length parameter of 𝐶^∗; we have 𝑓∶𝐼𝑡⟶⟶𝐼𝑡^∗_∗,=𝑓(𝑡)=||1+𝛽𝑘₁||𝑡.(3.46) Then, we obtain 𝑓^(𝑡)=√|1+𝛽𝑘₁| and 𝑓^(𝑡)𝐓^∗(𝑓(𝑡))==𝐓1(𝑡+)+𝛽𝑘𝛽₁𝐍^(𝑡)𝐓(𝑡)+𝛽𝑘₂𝐁₁(𝑡),(3.47) that is, 𝐓^∗(𝑓(𝑡))=||1+𝛽𝑘₁||𝐓(𝑡)+𝛽𝑘²||1+𝛽𝑘₁||𝐁¹(𝑡).(3.48) By differentiating the both sides of the above equality with respect to 𝑡, we find 𝑓^(𝑡)𝑑𝐓^∗𝑡^∗𝑑𝑡^∗||||_{𝑡∗=𝑓(𝑡)}=||1+𝛽𝑘₁||𝐓^(𝑡)+𝛽𝑘²||1+𝛽𝑘₁||𝐁^1=⎡⎢⎢⎢⎣𝑘(𝑡)₁1+𝛽𝑘₁−𝛽𝑘²²||1+𝛽𝑘₁||⎤⎥⎥⎥⎦𝐍(𝑡)+⎡⎢⎢⎢⎣𝛽𝑘²𝑘₃(𝑡)||1+𝛽𝑘₁||⎤⎥⎥⎥⎦𝐁²=⎡⎢⎢⎢⎣𝛽𝑘₂(𝑡)𝑘₃(𝑡)||1+𝛽𝑘₁||⎤⎥⎥⎥⎦𝐁²(𝑡).(3.49) Hence, since 𝑘₃ does not vanish, we get 𝑘^∗1(𝑓(𝑡))=‖‖‖‖𝑑𝐓^∗𝑡^∗𝑑𝑡^∗||||_{𝑡∗=𝑓(𝑡)}‖‖‖‖=𝜀𝛽𝑘₂𝑘₃(𝑡)||1+𝛽𝑘₁||>0,(3.50) where 𝜀=sign(𝑘₃) denotes the sign of function 𝑘₃. That is, 𝜀 is −1 or +1.
We can put𝐍^∗𝑡^∗=1𝑘^∗1(𝑡^∗)𝑑𝐓^∗𝑡^∗𝑑𝑡^∗,𝑡∈𝐼.(3.51) Then, we get 𝐍^∗(𝑓(𝑡))=𝜀𝐁₂(𝑡).(3.52) Differentiating of the last equation with respect to 𝑡, we reach 𝑓^(𝑡)𝑑𝐍^∗𝑡^∗𝑑𝑡^∗||||_{𝑡∗=𝑓(𝑡)}=−𝜀𝑘₃𝐁₁(𝑡),(3.53) and we have 𝑑𝐍^∗𝑡^∗𝑑𝑡^∗||||_{𝑡∗=𝑓(𝑡)}−𝑘^∗1(𝑓(𝑡))𝐓^∗(𝑓(𝑡))=−𝜀𝛽𝑘²𝑘₃(𝑡)||1+𝛽𝑘₁||𝐓(𝑡)−𝜀||1+𝛽𝑘₁||𝑘₃(𝑡)𝐁₁(𝑡).(3.54) Since 𝜀𝑘₃(𝑡) is positive for 𝑡∈𝐼, we have 𝑘^∗2(𝑓(𝑡))=‖‖‖‖𝑑𝐍^∗𝑡^∗𝑑𝑡^∗||||_{𝑡∗=𝑓(𝑡)}=⎷−𝑘^∗1(𝑓(𝑡))𝐓^∗(𝑓(𝑡))‖‖‖‖|||||−𝛽₂𝑘²²𝑘₃(𝑡)²||1+𝛽𝑘₁||+||1+𝛽𝑘¹||𝑘₃(𝑡)²=|||||𝑘₃(𝑡)²=𝜀𝑘₃(𝑡)>0.(3.55) Thus, we can put 𝐁_1∗(𝑓(𝑡))=1𝑘^∗2(𝑓(𝑡))𝑑𝐍^∗𝑡^∗𝑑𝑡^∗||||_{𝑡∗=𝑓(𝑡)}=−𝛽𝑘₂−𝑘^∗1(𝑓(𝑡))𝐓^∗(𝑓(𝑡))||1+𝛽𝑘₁||𝐓(𝑡)−||1+𝛽𝑘₁||𝐁₁(𝑡),𝑡∈𝐼.(3.56) By differentiation of the above equation with respect to 𝑡, we get 𝑓^(𝑡)𝑑𝐁^1∗𝑡^∗𝑑𝑡^∗||||_{𝑡∗=𝑓(𝑡)}=𝑘²||1+𝛽𝑘₁||𝐍(𝑡)−𝑘³(𝑡)||1+𝛽𝑘₁||𝐁₂(𝑡).(3.57) Since 𝑓^(𝑡)=√|1+𝛽𝑘₁| and 𝑘^∗2(𝑓(𝑡))𝐍^∗(𝑓(𝑡))=𝑘₃(𝑡)𝐁₂(𝑡), we have 𝑑𝐁_1∗𝑡^∗𝑑𝑡^∗||||_{𝑡∗=𝑓(𝑡)}+𝑘^∗2(𝑓(𝑡))𝐍^∗(𝑓(𝑡))=𝑘²||1+𝛽𝑘₁||𝐍(𝑡).(3.58) Thus, we obtain 𝐁_2∗(𝑓(𝑡))=𝛿𝐍(𝑡) for 𝑡∈𝐼, where 𝛿=∓1. We must determine whether 𝛿 is −1 or +1 under the condition that the frame field {𝐓^∗(𝑡),𝐍^∗(𝑡),𝐁^∗1(𝑡),𝐁^∗2(𝑡)} is of positive orientation.
We have, by det[𝐓(𝑡),𝐍(𝑡),𝐁₁(𝑡),𝐁₂(𝑡)]=1 for 𝑡∈𝐼,det=𝐓^∗de(𝑡t)⎡⎢⎢⎢⎢⎢⎣,𝐍^∗(𝑡),𝐁^∗1(𝑡),𝐁^∗2(𝑡)||1+𝛽𝑘₁||𝐓(𝑡)+𝛽𝑘²||1+𝛽𝑘₁||𝐁¹𝜀𝐁₂(𝑡),(𝑡),−𝛽𝑘²||1+𝛽𝑘₁||𝐓(𝑡)−=𝜀𝛿||1+𝛽𝑘₁||1+𝛽𝑘₁||𝐁₁(𝑡),𝛿𝐍(𝑡)⎤⎥⎥⎥⎥⎥⎦||−𝛽²𝑘²²||1+𝛽𝑘₁||=𝜀𝛿(3.59) and det[𝐓^∗(𝑡),𝐍^∗(𝑡),𝐁^∗1(𝑡),𝐁^∗2(𝑡)]=1 for any 𝑡∈𝐼. Therefore, we get 𝜀=𝛿. Thus, we get 𝑘𝐁^∗2(𝑓(𝑡))=𝜀𝐍(𝑡),_∗3(𝑓(𝑡))=𝑑𝐁^∗1𝑡^∗𝑑𝑡^∗||||_{𝑡∗=𝑓(𝑡)}=𝜀𝑘₂,𝐁^∗2(𝑓(𝑡))||1+𝛽𝑘₁||,𝑡∈𝐼.(3.60) By the above facts, 𝐶^∗ is a special Frenet curve in 𝐸⁴¹ and the first normal line at each point of 𝐶 is the third normal line of 𝐶^∗ at corresponding each point under the bijection 𝜙∶𝐜(𝑡)→𝜙(𝐜(𝑡))=𝐜^∗(𝑓(𝑡))∈𝐶^∗.
Thus, the proof is completed.

The following theorem gives a parametric representation of a generalized timelike Mannheim curves 𝐸⁴¹.

Theorem 3.6. Let 𝐶 be a timelike special curve defined by 𝐜(𝑠)=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝛽𝛽𝛽𝛽𝑓𝑓𝑓𝑓((((𝑠𝑠𝑠𝑠))))csℎ𝑔oins((hh𝑠𝑠))𝑠𝑠𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠⎤⎥⎥⎥⎥⎥⎥⎥⎦,𝑠∈𝑈⊂ℝ.(3.61) Here, 𝛽 is a nonzero constant number, 𝑔∶𝑈→ℝ and ℎ∶𝑈→ℝ are any smooth functions, and the positive-valued smooth function 𝑓∶𝑈→ℝ is given by 𝑓=1−𝑔²(𝑠)−ℎ²(𝑠)^−3/21−𝑔²(𝑠)−ℎ²(𝑠)+𝑔̇²(𝑠)+ℎ̇²(𝑠)−𝑔̇(𝑠)ℎ(𝑠)−𝑔(𝑠)ℎ̇(𝑠)²^−5/2×⎡⎢⎢⎢⎣−1−𝑔²(𝑠)−ℎ²(𝑠)+𝑔̇²(𝑠)+ℎ̇²(𝑠)−𝑔̇(𝑠)ℎ(𝑠)−𝑔(𝑠)ℎ̇(𝑠)²³+1−𝑔²(𝑠)−ℎ²(𝑠)³⎛⎜⎜⎜⎝−(𝑔(𝑠)−𝑔̈(𝑠))²−ℎ(𝑠)−ℎ̈(𝑠)²−𝑔(𝑠)ℎ̇(𝑠)−ġ(𝑠)ℎ(𝑠)−𝑔̇(𝑠)ℎ̈(𝑠)−𝑔̈(𝑠)ℎ̇(𝑠)²+𝑔(𝑠)ℎ̈(𝑠)−𝑔̈(𝑠)ℎ(𝑠)²⎞⎟⎟⎟⎠⎤⎥⎥⎥⎦,(3.62) for 𝑠∈𝑈. Then, the curvature functions 𝑘₁ and 𝑘₂ of 𝐶 satisfy 𝑘₁=−𝛽𝑘²¹−𝑘²²(3.63) at each point 𝐜(𝑠) of 𝐶.

Proof. Let 𝐶 be a timelike special curve defined by 𝐜(𝑠)=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝛽𝛽𝛽𝛽𝑓𝑓𝑓𝑓((((𝑠𝑠𝑠𝑠))))csℎ𝑔oin(s(h𝑠𝑠h))𝑠𝑠𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠⎤⎥⎥⎥⎥⎥⎥⎥⎦,𝑠∈𝑈⊂ℝ,(3.64) where 𝛽 is a nonzero constant number 𝑔 and ℎ are any smooth functions. 𝑓 is a positive-valued smooth function. Thus, we obtain 𝐜̇(𝑠)=⎡⎢⎢⎢⎣𝛽𝛽𝛽𝛽𝑓𝑓𝑓𝑓((((𝑠𝑠𝑠𝑠))))csℎ𝑔oins((hh𝑠𝑠))𝑠𝑠⎤⎥⎥⎥⎦,𝑠∈𝑈⊂ℝ,(3.65) where the subscript prime (⋅) denotes the differentiation with respect to 𝑠.
The arc-length parameter 𝑡 of 𝐶 is given by𝑡=𝜓(𝑠)=^𝑠𝑠₀‖𝐜̇(𝑠)‖𝑑𝑠,(3.66) where ‖𝐜(̇𝑠)‖=𝛽𝑓(𝑠)√−1+𝑔²(𝑠)+ℎ²(𝑠).
If 𝜑 denotes the inverse function of 𝜓∶𝑈→𝐼⊂ℝ, then 𝑠=𝜑(𝑡) and we get𝜑^(𝑡)=‖‖‖𝑑𝐜(𝑠)𝑑𝑠|||_{𝑠=𝜑(𝑡)}‖‖‖⁻¹,𝑡∈𝐼,(3.67) where the prime (^) denotes the differentiation with respect to 𝑡.
The unit tangent vector 𝐓(𝑡) of the curve 𝐶 at the each point 𝐜(𝜑(𝑡)) is given by𝐓(𝑡)=−1+𝑔²(𝜑(𝑡))+ℎ²(𝜑(𝑡))^−1/2⎡⎢⎢⎢⎣csoiℎ𝑔nshh((𝜑𝜑((𝜑𝜑((𝑡𝑡)()()𝑡)𝑡))))⎤⎥⎥⎥⎦,𝑡∈𝐼.(3.68) Some simplifying assumptions are made for the sake of brevity as follows: sinh∶=sinh(𝜑(𝑡)),cosh∶=cosh(𝜑(𝑡)),𝑔̇∶=𝑓𝑔̇∶(𝜑=(𝑓𝑡))(𝜑=(𝑡𝑑))𝑔,(𝑠)𝑔∶=𝑔(𝜑(𝑡)),ℎ∶=ℎ(𝜑(𝑡)),𝑑𝑠|||_{𝑠=𝜑(𝑡)},ℎ̇∶=ℎ̇(𝜑(𝑡))=𝑑ℎ(𝑠)𝑑𝑠|||_{𝑠=𝜑(𝑡)}𝑔̈∶=𝑔̈(𝜑(𝑡))=𝑑₂,𝑔(𝑠)𝑑𝑠²||||_{𝑠=𝜑(𝑡)},ℎ̈∶=ℎ̈(𝜑(𝑡))=𝑑²ℎ(𝑠)𝑑𝑠²||||_{𝑠=𝜑(𝑡)}𝜑_,∶=𝜑^(𝑡)=𝑑𝜑𝑑𝑡|||_𝑡𝐴∶=1−𝑔₂,𝐷∶=−𝑔𝑔̈−−ℎℎℎ̈,²,𝐸𝐵∶=∶=−−𝑔̇𝑔𝑔̈𝑔−̇−ℎ̇ℎℎ̈ℎ,̇,𝐹𝐶∶=∶=𝑔̈₂−𝑔̇²−ℎ̇²,+ℎ̈².(3.69) Thus, we get 𝐴̇=2𝐵,𝐵̇=𝐶+𝐷,𝐶̇=2𝐸,𝜑^=𝛽⁻¹𝑓⁻¹𝐴^−1/2.(3.70) So, we rewrite (3.68) as 𝐓∶=𝐓(𝑡)=𝐴^−1/2⎡⎢⎢⎢⎣csoiℎ𝑔nshh⎤⎥⎥⎥⎦.(3.71) Differentiating the last equation with respect to 𝑡, we find 𝐓^=𝜑^⎡⎢⎢⎢⎢⎢⎢⎢⎣−12𝐴^−3/2𝐴̇cosh+𝐴^−1/2sinh−12𝐴^−3/2𝐴̇sinh+𝐴^−1/2cosh−12𝐴^−3/2−1𝐴𝑔̇+𝐴^−1/2𝑔̇2𝐴^−3/2Ȧ‌ℎ+𝐴^−1/2ℎ̇⎤⎥⎥⎥⎥⎥⎥⎥⎦,(3.72) that is, 𝐓^=−𝜑^𝐴^−1/2⎡⎢⎢⎢⎢⎣𝐴𝐴⁻¹𝐵cosh−sinh₋₁𝐴𝐵sinh−cosh₋₁𝐴𝐵𝑔−𝑔₋₁̇𝐵ℎ−ℎ̇⎤⎥⎥⎥⎥⎦.(3.73) From the last equation, we find 𝑘₁∶=𝑘₁(𝑡)=‖‖𝐓^(𝑡)‖‖=𝜑^𝐴⁻¹𝐴−𝐴𝐶+𝐵²^1/2.(3.74) By the fact that 𝐍(𝑡)=(𝑘₁(𝑡))⁻¹𝐓^(𝑡), we get 𝐍∶=𝐍(𝑡)=−𝐴^1/2𝐴−𝐴𝐶+𝐵²^−1/2⎡⎢⎢⎢⎢⎣𝐴𝐴⁻¹𝐵cosh−sinh₋₁𝐴𝐵sinh−cosh₋₁𝐴𝐵𝑔−𝑔₋₁̇𝐵ℎ−ℎ̇⎤⎥⎥⎥⎥⎦.(3.75) In order to get second curvature function 𝑘₂, we need to calculate 𝑘₂(𝑡)=‖𝐍^(𝑡)−𝑘₁(𝑡)𝐓(𝑡)‖. After a long process of calculations and using abbreviations, we obtain 𝐍^−𝑘₁𝐓=𝜑^𝐴^−3/2𝐴−𝐴𝐶+𝐵²^−3/2⎡⎢⎢⎢⎢⎣((𝑃𝑃++𝑃𝑃𝑄𝑄ℎ𝑔))−−csio𝑅𝑅nshℎ𝑔ḣ̇++−−𝑄𝑄𝑅𝑅𝑔ℎcs̈̈oinshh⎤⎥⎥⎥⎥⎦,(3.76) where 𝑃=𝐴−𝐴𝐶+𝐵²𝐵²−𝐴𝑄𝐶=−𝐴𝐴₂𝐷−𝐴−𝐴𝐶+𝐵²²+𝐴𝐵(𝐵−𝐴𝐸+𝐵𝐷),𝑅=𝐴₂𝐴−𝐴𝐶+𝐵²,(𝐵−𝐴𝐸+𝐵𝐷).(3.77) If we simplify 𝑃, then we have 𝑃=𝐴²(𝐶−𝐵𝐸−𝐷+𝐶𝐷−1).(3.78) Therefore, we rewrite (3.76) and (3.77) as 𝐍^−𝑘₁𝐓=𝜑^𝐴^−1/2𝐴−𝐴𝐶+𝐵²^−3/2⎡⎢⎢⎢⎢⎢⎣𝑃𝑃++𝑃𝑃𝑄𝑄ℎ𝑔−−cs𝑅𝑅ionsℎ𝑔hḣ̇++−−𝑄𝑄𝑅𝑅𝑔ℎc̈̈soinshh⎤⎥⎥⎥⎥⎥⎦,(3.79) where 𝑃=𝑄𝐶−=𝐷𝐴+−𝐶𝐴𝐷𝐶−+𝐵𝐵𝐸₂−1,𝑅=𝐵−𝐴𝐸+𝐵𝐷,.(3.80) Consequently, from (3.79) and (3.80), we have ‖‖𝐍^−𝑘₁𝐓‖‖²=𝜑^²𝐴𝐴−𝐴𝐶+𝐵²⁻³⎡⎢⎢⎢⎢⎣−𝑃+𝑄+𝑄²₂+𝑅²+𝑃²𝑔²+ℎ²+𝑅²𝑔̇²+ℎ̇²−2𝑅𝑄𝑔𝑔̈²̇𝑔+̈+ℎℎ̈²̇ℎ̈−+2𝑃2𝑃𝑅𝑄𝑔𝑔𝑔̇𝑔+̈+ℎℎℎ̇ℎ̈⎤⎥⎥⎥⎥⎦.(3.81) Substituting the abbreviations into the last equation, we have ‖‖𝐍^−𝑘₁𝐓‖‖²=𝜑^²𝐴𝐴−𝐴𝐶+𝐵²⁻³×−𝑃²𝐴−2𝑃𝑄−𝑄²+𝑅²−𝑅²𝐶+𝑄²𝐹+2𝑃𝑅𝐵+2𝑅𝑄𝐸−2𝑃𝑄𝐷.(3.82) After substituting (3.80) into the last equation and simplifying it, we get 𝑘²²=‖‖𝐍^−𝑘₁𝐓‖‖²=𝜑^²𝐴𝐴−𝐴𝐶+𝐵²⁻²𝐴−𝐴𝐶+𝐵²(1−𝐹)+(𝐶−1)(1+𝐷)²−2𝐵𝐸(1+𝐷)+𝐴𝐸².(3.83) Moreover, from (3.74) it is seen that 𝑘²¹=𝜑^²𝐴⁻²𝐴−𝐴𝐶+𝐵².(3.84) The last two equations show us that 𝑘²²−𝑘²¹=𝜑^²𝐴⁻²𝐴−𝐴𝐶+𝐵²⁻²×−𝐴−𝐴𝐶+𝐵²³+𝐴³𝐴−𝐴𝐶+𝐵²(1−𝐹)+(𝐶−1)(1+𝐷)²−2𝐵𝐸(1+𝐷)+𝐴𝐸².(3.85) By the fact 𝜑^=𝛽⁻¹𝑓⁻¹𝐴^−1/2, we obtain 𝑘²²−𝑘²¹=𝛽⁻²𝑓⁻²𝐴⁻³𝐴−𝐴𝐶+𝐵²⁻²×𝐴−𝐴𝐶+𝐵²³+𝐴³𝐴−𝐴𝐶+𝐵²(1−𝐹)+(𝐶−1)(1+𝐷)²−2𝐵𝐸(1+𝐷)+𝐴𝐸²,(3.86)𝑘₁=𝛽⁻¹𝑓⁻¹𝐴^−3/2𝐴−𝐴𝐶+𝐵²^1/2.(3.87) According to our assumption 𝑓=1−𝑔²−ℎ²^−3/21−𝑔²−ℎ²+𝑔̇²+ℎ̇²−𝑔̇ℎ−𝑔ℎ̇²^−5/2×−1−𝑔²−ℎ²+𝑔̇²+ℎ̇²−𝑔̇ℎ−𝑔ℎ̇²³+1−𝑔²−ℎ²³−(𝑔−𝑔̈)²−ℎ−ℎ̈²−𝑔ℎ̇−𝑔̇ℎ−𝑔̇ℎ̈−𝑔̈ℎ̇²+𝑔ℎ̈−𝑔̈ℎ²,(3.88) we obtain 𝑓=𝐴^−3/2𝐴−𝐴𝐶+𝐵²^−5/2⎡⎢⎢⎢⎣𝐴−𝐴𝐶+𝐵²³+𝐴³𝐴−𝐴𝐶+𝐵²(1−𝐹)+(𝐶−1)(1+𝐷)²−2𝐵𝐸(1+𝐷)+𝐴𝐸²⎤⎥⎥⎥⎦.(3.89) Substituting the above equation into (3.86) and (3.87), we obtain 𝑘₁=−𝛽𝑘²¹−𝑘²².(3.90) The proof is completed.