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Mathematical Problems in Engineering
VolumeΒ 2011, Article IDΒ 539378, 19 pages
Research Article

Generalized Timelike Mannheim Curves in Minkowski Space-Time πΈπŸ’πŸ

Department of Mathematics, Faculty of Arts and Sciences, Sakarya University, 54187 Sakarya, Turkey

Received 19 January 2011; Accepted 4 March 2011

Academic Editor: CarloΒ Cattani

Copyright Β© 2011 M. Akyig~it et al. 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.


We give the definition of generalized timelike Mannheim curve in Minkowski space-time πΈπŸ’πŸ. 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 π‘˜1 and torsion π‘˜2 satisfy the relationπ‘˜1ξ€·π‘˜=𝛽21+π‘˜22ξ€Έ(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 ⟨,⟩=βˆ’π‘‘π‘₯21+𝑑π‘₯22+𝑑π‘₯23+𝑑π‘₯24,(2.1) where (π‘₯1,π‘₯2,π‘₯3,π‘₯4) 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(𝑑) in 𝐸41 can locally be spacelike, timelike, or null (lightlike) if all of its velocity vectors πœξ…ž(𝑑) are, respectively, spacelike, timelike, or null. The norm of 𝐯∈𝐸41 is given by βˆšβ€–π―β€–=|⟨𝐯,𝐯⟩|. If β€–πœξ…žβˆš(𝑑)β€–=|βŸ¨πœξ…ž(𝑑),πœξ…ž(𝑑)⟩|β‰ 0 for all π‘‘βˆˆπΌ, then 𝐢 is a regular curve in 𝐸41. A timelike (spacelike) regular curve 𝐢 is parameterized by arc-length parameter 𝑑 which is given by πœβˆΆπΌβ†’πΈ41, then the tangent vector πœξ…ž(𝑑) along 𝐢 has unit length, that is, βŸ¨πœξ…ž(𝑑),πœξ…ž(𝑑)⟩=βˆ’1, (βŸ¨πœξ…ž(𝑑),πœξ…ž(𝑑)⟩=1) for all π‘‘βˆˆπΌ.

Hereafter, curves considered are timelike and regular 𝐢∞ curves in 𝐸41. Let 𝐓(𝑑)=πœξ…ž(𝑑) for all π‘‘βˆˆπΌ; then the vector field 𝐓(𝑑) is timelike and it is called timelike unit tangent vector field on 𝐢.

The timelike curve 𝐢 is called special timelike Frenet curve if there exist three smooth functions π‘˜1, π‘˜2, π‘˜3 on 𝐢 and smooth nonnull frame field {𝐓,𝐍,𝐁1,𝐁2} along the curve 𝐢. Also, the functions π‘˜1, π‘˜2, and π‘˜3 are called the first, the second, and the third curvature function on 𝐢, respectively. For the 𝐢∞ special timelike Frenet curve 𝐢, the following Frenet formula isβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π“ξ…žπξ…žπξ…ž1πξ…ž2⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=⎑⎒⎒⎒⎣0π‘˜1π‘˜0010π‘˜200βˆ’π‘˜20π‘˜300βˆ’π‘˜30⎀βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ£π“ππ1𝐁2⎀βŽ₯βŽ₯βŽ₯⎦,(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 βŸ¨π“,π“βŸ©=βˆ’1,⟨𝐍,𝐍⟩=⟨𝐁1,𝐁1⟩=⟨𝐁2,𝐁2⟩=1.(2.3) For π‘‘βˆˆπΌ, the nonnull frame field {𝐓,𝐍,𝐁1,𝐁2} and curvature functions π‘˜1, π‘˜2, and π‘˜3 are determined as follows:1ststep𝐓(𝑑)=πœξ…ž2(𝑑),ndstepπ‘˜1‖‖𝐓(𝑑)=ξ…žβ€–β€–1(𝑑)>0,𝐍(𝑑)=π‘˜1𝐓(𝑑)ξ…ž(𝑑),3rdstepπ‘˜2‖‖𝐍(𝑑)=ξ…ž(𝑑)βˆ’π‘˜1‖‖𝐁(𝑑)𝐓(𝑑)>0,1(1𝑑)=π‘˜2𝐍(𝑑)ξ…ž(𝑑)βˆ’π‘˜1(ξ€Έ,4𝑑)𝐓(𝑑)thstep𝐁21(𝑑)=π›Ώβ€–β€–πξ…ž1(𝑑)+π‘˜2‖‖𝐁(𝑑)𝐍(𝑑)ξ…ž1(𝑑)+π‘˜2ξ€Έ,(𝑑)𝐍(𝑑)(2.4) where 𝛿 is determined by the fact that orthonormal frame field {𝐓(𝑑),𝐍(𝑑),𝐁1(𝑑),𝐁2(𝑑)} is of positive orientation. The function π‘˜3 is determined byπ‘˜3𝐁(𝑑)=ξ…ž1(𝑑),𝐁2(𝑑)β‰ 0.(2.5) So the function π‘˜3 never vanishes.

In order to make sure that the curve 𝐢 is a special timelike Frenet curve, above steps must be checked, from 1st step to 4th step, for π‘‘βˆˆπΌ.

Let {𝐓,𝐍,𝐁1,𝐁2} be the moving Frenet frame along a unit speed timelike curve 𝐢 in 𝐸41, consisting of the tangent, the principal normal, the first binormal, and the second binormal vector field, respectively. Since 𝐢 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 𝐢 in 𝐸41 is a generalized timelike Mannheim curve if there exists a special timelike Frenet curve πΆβˆ— in 𝐸41 such that the first normal line at each point of 𝐢 is included in the plane generated by the second normal line and the third normal line of πΆβˆ— at the corresponding point under πœ™. Here, πœ™ is a bijection from 𝐢 to πΆβˆ—. The curve πΆβˆ— is called the generalized timelike Mannheim mate curve of 𝐢.

By the definition, a generalized Mannheim mate curve πΆβˆ— is given by the map πœβˆ—βˆΆπΌβˆ—β†’πΈ41 such thatπœβˆ—(𝑑)=𝐜(𝑑)+𝛽(𝑑)𝐍(𝑑),π‘‘βˆˆπΌ.(3.1) Here 𝛽 is a smooth function on 𝐼. Generally, the parameter 𝑑 is not an arc-length of πΆβˆ—. Let π‘‘βˆ— be the arc-length of πΆβˆ— defined by π‘‘βˆ—=ξ€œπ‘‘0β€–β€–β€–π‘‘πœβˆ—(𝑑)‖‖‖𝑑𝑑𝑑𝑑.(3.2) If a smooth function π‘“βˆΆπΌβ†’πΌβˆ— is given by 𝑓(𝑑)=π‘‘βˆ—, then for all π‘‘βˆˆπΌ, we have π‘“ξ…ž(𝑑)=π‘‘π‘‘βˆ—=β€–β€–β€–π‘‘π‘‘π‘‘πœβˆ—(𝑑)β€–β€–β€–=𝑑𝑑|||βˆ’ξ€·1+𝛽(𝑑)π‘˜1ξ€Έ(𝑑)2+ξ€·π›½ξ…žξ€Έ(𝑑)2+𝛽(𝑑)π‘˜2ξ€Έ(𝑑)2|||.(3.3) The representation of timelike curve πΆβˆ— with arc-length parameter π‘‘βˆ— is πœβˆ—βˆΆπΌβˆ—βŸΆπΈ41,π‘‘βˆ—βŸΆπœβˆ—ξ€·π‘‘βˆ—ξ€Έ.(3.4) For a bijection πœ™βˆΆπΆβ†’πΆβˆ— defined by πœ™(𝐜(𝑑))=πœβˆ—(𝑓(𝑑)), the reparameterization of πΆβˆ— isπœβˆ—(𝑓(𝑑))=𝐜(𝑑)+𝛽(𝑑)𝐍(𝑑),(3.5) where 𝛽 is a smooth function on 𝐼. Thus, we have π‘‘πœβˆ—(𝑓(𝑑))=π‘‘π‘‘π‘‘πœβˆ—ξ€·π‘‘βˆ—ξ€Έ||||π‘‘π‘‘π‘‘βˆ—=𝑓(𝑑)π‘“ξ…ž(𝑑)=π‘“ξ…ž(𝑑)π“βˆ—(𝑓(𝑑)),π‘‘βˆˆπΌ.(3.6)

Theorem 3.2. If a special timelike Frenet curve 𝐢 in 𝐸41 is a generalized timelike Mannheim curve, then the following relation between the first curvature function π‘˜1 and the second curvature function π‘˜2 holds: π‘˜1ξ€·π‘˜(𝑑)=βˆ’π›½21(𝑑)βˆ’π‘˜22ξ€Έ(𝑑),π‘‘βˆˆπΌ,(3.7) where 𝛽 is a constant number.

Proof. Let 𝐢 be a generalized timelike Mannheim curve and πΆβˆ— the generalized timelike Mannheim mate curve of 𝐢, as the following diagram: raouf(3.8) A smooth function 𝑓 is defined by βˆ«π‘“(𝑑)=β€–π‘‘πœβˆ—(𝑑)/𝑑𝑑‖𝑑𝑑=π‘‘βˆ— and π‘‘βˆ— is the arc-length parameter of πΆβˆ—. Also, πœ™ is a bijection defined by πœ™(𝐜(𝑑))=πœβˆ—(𝑓(𝑑)). Thus, the timelike curve πΆβˆ— is reparametrized as follows πœβˆ—(𝑓(𝑑))=𝐜(𝑑)+𝛽(𝑑)𝐍(𝑑),(3.9) where π›½βˆΆπΌβŠ‚β„β†’β„ is a smooth function. By differentiating both sides of (3.9) with respect to 𝑑, we have π‘“ξ…ž(𝑑)π“βˆ—ξ€·(𝑓(𝑑))=1+𝛽(𝑑)π‘˜1ξ€Έ(𝑑)𝐓+π›½ξ…ž(𝑑)𝐍(𝑑)+𝛽(𝑑)π‘˜2(𝑑)𝐁1(𝑑).(3.10) On the other hand, since the first normal line at each point of 𝐢 is lying in the plane generated by the second normal line and the third normal line of πΆβˆ— at the corresponding points under bijection πœ™, the vector field 𝐍(𝑑) is given by 𝐍(𝑑)=𝑔(𝑑)𝐁1βˆ—(𝑓(𝑑))+β„Ž(𝑑)𝐁2βˆ—,(𝑓(𝑑))(3.11) where 𝑔 and β„Ž are some smooth functions on πΌβŠ‚β„. If we take into consideration βŸ¨π“βˆ—(𝑓(𝑑)),𝑔(𝑑)𝐁1βˆ—(𝑓(𝑑))+β„Ž(𝑑)𝐁2βˆ—(𝑓(𝑑))⟩=0(3.12) and (3.10), then we have π›½ξ…ž(𝑑)=0. So we rewrite (3.10) as π‘“ξ…ž(𝑑)π“βˆ—ξ€·(𝑓(𝑑))=1+π›½π‘˜1𝐓(𝑑)(𝑑)+π›½π‘˜2(𝑑)𝐁1(𝑑),(3.13) that is, π“βˆ—ξ€·(𝑓(𝑑))=1+π›½π‘˜1ξ€Έ(𝑑)π‘“ξ…ž(𝑑)𝐓(𝑑)+π›½π‘˜2(𝑑)π‘“ξ…žπ(𝑑)1(𝑑),(3.14) where π‘“ξ…žξ‚™(𝑑)=|||βˆ’ξ€·1+π›½π‘˜1ξ€Έ(𝑑)2+ξ€·π›½π‘˜2ξ€Έ(𝑑)2|||.(3.15) By taking the differentiations both sides of (3.13) with respect to π‘‘βˆˆπΌ, we get π‘“ξ…ž(𝑑)π‘˜βˆ—1(𝑓(𝑑))πβˆ—ξ‚΅(𝑓(𝑑))=1+π›½π‘˜1(𝑑)π‘“ξ…žξ‚Ά(𝑑)ξ…ž+𝐓(𝑑)1+π›½π‘˜1ξ€Έπ‘˜(𝑑)1ξ€·π‘˜(𝑑)βˆ’π›½2ξ€Έ(𝑑)2π‘“ξ…ž(ξƒͺ+𝑑)𝐍(𝑑)π›½π‘˜2(𝑑)π‘“ξ…žξ‚Ά(𝑑)ξ…žπ1ξ‚΅(𝑑)+π›½π‘˜2(𝑑)π‘˜3(𝑑)π‘“ξ…žξ‚Άπ(𝑑)2(𝑑).(3.16) Since βŸ¨πβˆ—(𝑓(𝑑)),𝑔(𝑑)𝐁1βˆ—(𝑓(𝑑))+β„Ž(𝑑)𝐁2βˆ—(𝑓(𝑑))⟩=0,(3.17) the coefficient of 𝐍(𝑑) in (3.16) vanishes, that is, ξ€·1+π›½π‘˜1ξ€Έπ‘˜(𝑑)1ξ€·π‘˜(𝑑)βˆ’π›½2ξ€Έ(𝑑)2=0.(3.18) Thus, this completes the proof.

Theorem 3.3. In 𝐸41, let 𝐢 be a special timelike Frenet curve such that its nonconstant first and second curvature functions satisfy the equality π‘˜1(𝑠)=βˆ’π›½(π‘˜21(𝑑)βˆ’π‘˜22(𝑑)) for all π‘‘βˆˆπΌβŠ‚β„. If the timelike curve πΆβˆ— given by πœβˆ—(𝑑)=𝐜(𝑑)+𝛽𝐍(𝑑)(3.19) is a special timelike Frenet curve, then πΆβˆ— is a generalized timelike Mannheim mate curve of 𝐢.

Proof. The arc-length parameter of πΆβˆ— is given by π‘‘βˆ—=ξ€œπ‘‘0β€–β€–β€–π‘‘πœβˆ—(𝑑)‖‖‖𝑑𝑑𝑑𝑑,π‘‘βˆˆπΌ.(3.20) Under the assumption of π‘˜1ξ€·π‘˜(𝑑)=βˆ’π›½21(𝑑)βˆ’π‘˜22ξ€Έ,(𝑑)(3.21) we obtain π‘“ξ…žβˆš(𝑑)=|1+π›½π‘˜1(𝑑)|, π‘‘βˆˆπΌ.
Differentiating the equation πœβˆ—(𝑓(𝑑))=𝐜(𝑑)+𝛽𝐍(𝑑) with respect to 𝑑, we reachπ‘“ξ…ž(𝑑)π“βˆ—ξ€·(𝑓(𝑑))=1+π›½π‘˜1𝐓(𝑑)(𝑑)+π›½π‘˜2(𝑑)𝐁1(𝑑).(3.22) Thus, it is seen that π“βˆ—βŽ›βŽœβŽœβŽœβŽ(𝑓(𝑑))=1+π›½π‘˜1(𝑑)||1+π›½π‘˜1||(𝑑)𝐓(𝑑)+π›½π‘˜2(𝑑)||1+π›½π‘˜1||𝐁(𝑑)1⎞⎟⎟⎟⎠(𝑑),π‘‘βˆˆπΌ.(3.23) The differentiation of the last equation with respect to 𝑑 is π‘“ξ…ž(𝑑)π‘˜βˆ—1(𝑓(𝑑))πβˆ—ξ‚΅ξ”(𝑓(𝑑))=||1+π›½π‘˜1||ξ‚Ά(𝑑)ξ…ž+βŽ›βŽœβŽœβŽœβŽξ€·π“(𝑑)1+π›½π‘˜1ξ€Έπ‘˜(𝑑)1(𝑑)βˆ’π›½π‘˜22(𝑑)||1+π›½π‘˜1||⎞⎟⎟⎟⎠+βŽ›βŽœβŽœβŽœβŽ(𝑑)𝐍(𝑑)π›½π‘˜2(𝑑)||1+π›½π‘˜1||⎞⎟⎟⎟⎠(𝑑)ξ…žπ1βŽ›βŽœβŽœβŽœβŽ(𝑑)+π›½π‘˜2(𝑑)π‘˜3(𝑑)||1+π›½π‘˜1||⎞⎟⎟⎟⎠𝐁(𝑑)2(𝑑).(3.24) From our assumption, we have π‘˜1(𝑑)+π›½π‘˜21(𝑑)βˆ’π›½π‘˜22(𝑑)||1+π›½π‘˜1||(𝑑)=0.(3.25) Thus, the coefficient of 𝐍(𝑑) in (3.24) is zero. It is seen from (3.23) that π“βˆ—(𝑓(𝑑)) is a linear combination of 𝐓(𝑑) and 𝐁1(𝑑). Additionally, from (3.24), πβˆ—(𝑓(𝑑)) is given by linear combination of 𝐓(𝑑), 𝐁1(𝑑), and 𝐁2(𝑑). On the other hand, πΆβˆ— is a special timelike Frenet curve that the vector 𝐍(𝑑) is given by linear combination of πβˆ—1(𝑓(𝑑)) and πβˆ—2(𝑓(𝑑)).
Therefore, the first normal line of 𝐢 lies in the plane generated by the second normal line and third normal line of πΆβˆ— at the corresponding points under a bijection πœ™ defined by πœ™(𝐜(𝑑))=πœβˆ—(𝑓(𝑑)).
This completes the proof.

Remark 3.4. In 4-dimensional Minkowski space 𝐸41, a special timelike Frenet curve 𝐢 with curvature functions π‘˜1 and π‘˜2 satisfying π‘˜1(𝑑)=βˆ’π›½(π‘˜21(𝑑)βˆ’π‘˜22(𝑑)), it is not clear that a smooth timelike curve πΆβˆ— 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 𝐢 be a special timelike curve in 𝐸41 with nonzero third curvature function π‘˜3. There exists a timelike special Frenet curve πΆβˆ— in 𝐸41 such that the first normal line of 𝐢 is linearly dependent with the third normal line of πΆβˆ— at the corresponding points 𝐜(𝑑) and πœβˆ—(𝑑), respectively, under a bijection πœ™βˆΆπΆβ†’πΆβˆ—, if and only if the curvatures π‘˜1 and π‘˜2 of 𝐢 are constant functions.

Proof. Let 𝐢 be a timelike Frenet curve in 𝐸41 with the Frenet frame field {𝐓,𝐍,𝐁1,𝐁2} and curvature functions π‘˜1, π‘˜2, and π‘˜3. Also, we assume that πΆβˆ— is a timelike special Frenet curve in 𝐸41 with the Frenet frame field {π“βˆ—,πβˆ—,𝐁1βˆ—,𝐁2βˆ—} and curvature functions π‘˜βˆ—1, π‘˜βˆ—2, and π‘˜βˆ—3. Let the first normal line of 𝐢 be linearly dependent with the third normal line of πΆβˆ— at the corresponding points 𝐢 and πΆβˆ—, respectively. Then the parameterization of πΆβˆ— is πœβˆ—(𝑓(𝑑))=𝐜(𝑑)+𝛽(𝑑)𝐍(𝑑),π‘‘βˆˆπΌ.(3.26) If the arc-length parameter of πΆβˆ— is given π‘‘βˆ—, then π‘‘βˆ—=ξ€œπ‘‘0ξ‚™|||βˆ’ξ€·1+𝛽(𝑑)π‘˜1ξ€Έ(𝑑)2+ξ€·π›½ξ…žξ€Έ(𝑑)2+𝛽(𝑑)π‘˜2ξ€Έ(𝑑)2|||𝑑𝑑,(3.27)π‘“βˆΆπΌβŸΆπΌβˆ—,π‘‘βŸΆπ‘“(𝑑)=π‘‘βˆ—.(3.28) Moreover, πœ™βˆΆπΆβ†’πΆβˆ— is a bijection given by πœ™(𝐜(𝑑))=πœβˆ—(𝑓(𝑑)).
Differentiating (3.26) with respect to 𝑑 and using the Frenet formulas, we getπ‘“ξ…ž(𝑑)π“βˆ—ξ€·(𝑓(𝑑))=1+𝛽(𝑑)π‘˜1𝐓(𝑑)(𝑑)+π›½ξ…ž(𝑑)𝐍(𝑑)+𝛽(𝑑)π‘˜2(𝑑)𝐁1(𝑑).(3.29) Since 𝐁2βˆ—(𝑓(𝑑))=βˆ“π(𝑑), then ξ«π‘“ξ…ž(𝑑)π“βˆ—(𝑓(𝑑)),𝐁2βˆ—ξ¬ξ€·(𝑓(𝑑))=⟨1+𝛽(𝑑)π‘˜1𝐓(𝑑)(𝑑)+π›½ξ…ž(𝑑)𝐍(𝑑)+𝛽(𝑑)π‘˜2(𝑑)𝐁1(𝑑),βˆ“π(𝑑)⟩(3.30) that is, 0=βˆ“π›½ξ…ž(𝑑).(3.31) From the last equation, it is easily seen that 𝛽 is a constant number. Hereafter, we can denote 𝛽(𝑑)=𝛽, for all π‘‘βˆˆπΌ.
From (3.27), we haveπ‘“ξ…žξ‚™(𝑑)=|||βˆ’ξ€·1+π›½π‘˜1ξ€Έ(𝑑)2+ξ€·π›½π‘˜2ξ€Έ(𝑑)2|||>0.(3.32) Thus, we rewrite (3.29) as follows: π“βˆ—ξ‚΅(𝑓(𝑑))=1+π›½π‘˜1(𝑑)π‘“ξ…žξ‚Άξ‚΅(𝑑)𝐓(𝑑)+π›½π‘˜2(𝑑)π‘“ξ…žξ‚Άπ(𝑑)1(𝑑).(3.33) The differentiation of the last equation with respect to 𝑑 is π‘“ξ…ž(𝑑)π‘˜βˆ—1(𝑓(𝑑))πβˆ—ξ‚΅(𝑓(𝑑))=1+π›½π‘˜1(𝑑)π‘“ξ…žξ‚Ά(𝑑)ξ…ž+𝐓(𝑑)1+π›½π‘˜1ξ€Έπ‘˜(𝑑)1(𝑑)βˆ’π›½π‘˜22(𝑑)π‘“ξ…žξƒͺ𝐍+ξ‚΅(𝑑)(𝑑)π›½π‘˜2(𝑑)π‘“ξ…žξ‚Ά(𝑑)ξ…žπ1ξ‚΅(𝑑)+π›½π‘˜2(𝑑)π‘˜3(𝑑)π‘“ξ…žξ‚Άπ(𝑑)2(𝑑).(3.34) Since βŸ¨π‘“ξ…ž(𝑑)π‘˜βˆ—1(𝑓(𝑑))πβˆ—(𝑓(𝑑)),𝐁2βˆ—(𝑓(𝑑))⟩=0 and 𝐁2βˆ—(𝑓(𝑑))=βˆ“π(𝑑) for all π‘‘βˆˆπΌ, π‘˜1(𝑑)+π›½π‘˜21(𝑑)βˆ’π›½π‘˜22(𝑑)=0(3.35) is satisfied. Then π‘˜π›½=βˆ’1(𝑑)π‘˜21(𝑑)βˆ’π‘˜22(𝑑)(3.36) is a nonzero constant number. Thus, from (3.34), we reach πβˆ—1(𝑓(𝑑))=π‘“ξ…žξ‚΅(𝑑)𝐾(𝑑)1+π›½π‘˜1(𝑑)π‘“ξ…žξ‚Ά(𝑑)ξ…ž1𝐓(𝑑)+π‘“ξ…žξ‚΅(𝑑)𝐾(𝑑)π›½π‘˜2(𝑑)π‘“ξ…žξ‚Ά(𝑑)ξ…žπ1+1(𝑑)π‘“ξ…žξ‚΅(𝑑)𝐾(𝑑)π›½π‘˜2(𝑑)π‘˜3(𝑑)π‘“ξ…žξ‚Άπ(𝑑)2(𝑑),(3.37) where 𝐾(𝑑)=π‘˜βˆ—1(𝑓(𝑑)) for all π‘‘βˆˆπΌ. Differentiating the last equation with respect to 𝑑, then we have π‘“ξ…žξ€Ίπ‘˜(𝑑)βˆ—1(𝑓(𝑑))π“βˆ—(𝑓(𝑑))+π‘˜βˆ—2(𝑓(𝑑))πβˆ—1ξ€»=ξ‚΅1(𝑓(𝑑))π‘“ξ…žξ‚΅(𝑑)𝐾(𝑑)1+π›½π‘˜1(𝑑)π‘“ξ…žξ‚Ά(𝑑)ξ…žξ‚Άξ…žπ“+ξ‚΅π‘˜(𝑑)1(𝑑)π‘“ξ…žξ‚΅(𝑑)𝐾(𝑑)1+π›½π‘˜1(𝑑)π‘“ξ…žξ‚Ά(𝑑)ξ…žβˆ’π‘˜2(𝑑)π‘“ξ…žξ‚΅(𝑑)𝐾(𝑑)π›½π‘˜2(𝑑)π‘“ξ…žξ‚Ά(𝑑)ξ…žξ‚Ά+1𝐍(𝑑)π‘“ξ…žξ‚΅(𝑑)𝐾(𝑑)π›½π‘˜2(𝑑)π‘“ξ…žξ‚Ά(𝑑)ξ…žξ‚Άξ…žβˆ’π‘˜3(𝑑)π‘“ξ…žξ‚΅(𝑑)𝐾(𝑑)π›½π‘˜2(𝑑)π‘˜3(𝑑)π‘“ξ…žξ‚Άξƒͺ𝐁(𝑑)1(+1𝑑)ξ‚΅ξ‚΅π‘“ξ…ž(𝑑)𝐾(𝑑)π›½π‘˜2(𝑑)π‘˜3(𝑑)π‘“ξ…ž(𝑑)ξ‚Άξ‚Άξ…ž+π‘˜3(𝑑)π‘“ξ…ž(𝑑)𝐾(𝑑)π›½π‘˜2(𝑑)π‘“ξ…ž(𝑑)ξ…žξ‚Άπ2(𝑑)(3.38) for all π‘‘βˆˆπΌ. Considering ξ«π‘“ξ…žξ€·π‘˜(𝑑)βˆ—1(𝑓(𝑑))π“βˆ—(𝑓(𝑑))+π‘˜βˆ—2(𝑓(𝑑))πβˆ—1ξ€Έ(𝑓(𝑑)),πβˆ—2𝐁(𝑓(𝑑))=0,βˆ—2(𝑓(𝑑))=βˆ“π(𝑑),(3.39) then we get π‘˜1ξ‚΅(𝑑)1+π›½π‘˜1(𝑑)π‘“ξ…žξ‚Ά(𝑑)ξ…žβˆ’π‘˜2ξ‚΅(𝑑)π›½π‘˜2(𝑑)π‘“ξ…žξ‚Ά(𝑑)ξ…ž=0.(3.40) Arranging the last equation, we find π›½ξ€Ίπ‘˜1(𝑑)π‘˜ξ…ž1(𝑑)βˆ’π‘˜2(𝑑)π‘˜ξ…ž2𝑓(𝑑)ξ…žξ€Ίπ‘˜(𝑑)βˆ’1(𝑑)+π›½π‘˜21(𝑑)βˆ’π›½π‘˜22𝑓(𝑑)ξ…žξ…ž(𝑑)=0.(3.41) Moreover, the differentiation of (3.36) with respect to 𝑑 is π‘˜ξ…ž1ξ€·π‘˜(𝑑)+2𝛽1(𝑑)π‘˜ξ…ž1(𝑑)βˆ’π‘˜2(𝑑)π‘˜ξ…ž2ξ€Έ(𝑑)=0.(3.42) From the above equation, it is seen that βˆ’π‘˜ξ…ž1(𝑑)2ξ€·π‘˜=𝛽1(𝑑)π‘˜ξ…ž1(𝑑)βˆ’π‘˜2(𝑑)π‘˜ξ…ž2ξ€Έ.(𝑑)(3.43) Substituting (3.36) and (3.43) into (3.41), we obtain βˆ’π‘˜ξ…ž1(𝑑)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 π‘˜2 is positive constant, too.
Conversely, suppose that 𝐢 is a timelike Frenet curve in 𝐸41 with the Frenet frame field {𝐓,𝐍,𝐁1,𝐁2} and curvature functions π‘˜1, π‘˜2, and π‘˜3. The first curvature function π‘˜1 and the second curvature function π‘˜2 of 𝐢 are of positive constant. Thus, π‘˜1/(π‘˜22βˆ’π‘˜21) is a positive constant number, say 𝛽.
The representation of timelike curve πΆβˆ— with arc-length parameter 𝑑 isπœβˆ—βˆΆπΌβŸΆπΈ41,π‘‘βŸΆπœβˆ—(𝑑)=𝐜(𝑑)+𝛽(𝑑)𝐍(𝑑).(3.45) Let π‘‘βˆ— denote the arc-length parameter of πΆβˆ—; we have π‘“βˆΆπΌβŸΆπΌβˆ—,π‘‘βŸΆπ‘‘βˆ—ξ”=𝑓(𝑑)=||1+π›½π‘˜1||𝑑.(3.46) Then, we obtain π‘“ξ…žβˆš(𝑑)=|1+π›½π‘˜1| and π‘“ξ…ž(𝑑)π“βˆ—(𝑓(𝑑))=𝐓(𝑑)+π›½πξ…ž=ξ€·(𝑑)1+π›½π‘˜1𝐓(𝑑)+π›½π‘˜2𝐁1(𝑑),(3.47) that is, π“βˆ—ξ”(𝑓(𝑑))=||1+π›½π‘˜1||𝐓(𝑑)+π›½π‘˜2||1+π›½π‘˜1||𝐁1(𝑑).(3.48) By differentiating the both sides of the above equality with respect to 𝑑, we find π‘“ξ…ž(𝑑)π‘‘π“βˆ—ξ€·π‘‘βˆ—ξ€Έπ‘‘π‘‘βˆ—||||π‘‘βˆ—=𝑓(𝑑)=||1+π›½π‘˜1||π“ξ…ž(𝑑)+π›½π‘˜2||1+π›½π‘˜1||πξ…ž1=βŽ‘βŽ’βŽ’βŽ’βŽ£π‘˜(𝑑)1ξ€·1+π›½π‘˜1ξ€Έβˆ’π›½π‘˜22||1+π›½π‘˜1||⎀βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎣𝐍(𝑑)+π›½π‘˜2π‘˜3(𝑑)||1+π›½π‘˜1||⎀βŽ₯βŽ₯βŽ₯⎦𝐁2=⎑⎒⎒⎒⎣(𝑑)π›½π‘˜2π‘˜3(𝑑)||1+π›½π‘˜1||⎀βŽ₯βŽ₯βŽ₯⎦𝐁2(𝑑).(3.49) Hence, since π‘˜3 does not vanish, we get π‘˜βˆ—1β€–β€–β€–β€–(𝑓(𝑑))=π‘‘π“βˆ—ξ€·π‘‘βˆ—ξ€Έπ‘‘π‘‘βˆ—||||π‘‘βˆ—=𝑓(𝑑)β€–β€–β€–β€–=πœ€π›½π‘˜2π‘˜3(𝑑)||1+π›½π‘˜1||>0,(3.50) where πœ€=sign(π‘˜3) denotes the sign of function π‘˜3. That is, πœ€ is βˆ’1 or +1.
We can putπβˆ—ξ€·π‘‘βˆ—ξ€Έ=1π‘˜βˆ—1(π‘‘βˆ—)π‘‘π“βˆ—ξ€·π‘‘βˆ—ξ€Έπ‘‘π‘‘βˆ—,π‘‘βˆˆπΌ.(3.51) Then, we get πβˆ—(𝑓(𝑑))=πœ€π2(𝑑).(3.52) Differentiating of the last equation with respect to 𝑑, we reach π‘“ξ…ž(𝑑)π‘‘πβˆ—ξ€·π‘‘βˆ—ξ€Έπ‘‘π‘‘βˆ—||||π‘‘βˆ—=𝑓(𝑑)=βˆ’πœ€π‘˜3𝐁1(𝑑),(3.53) and we have π‘‘πβˆ—ξ€·π‘‘βˆ—ξ€Έπ‘‘π‘‘βˆ—||||π‘‘βˆ—=𝑓(𝑑)βˆ’π‘˜βˆ—1(𝑓(𝑑))π“βˆ—(𝑓(𝑑))=βˆ’πœ€π›½π‘˜2π‘˜3(𝑑)||1+π›½π‘˜1||𝐓(𝑑)βˆ’πœ€||1+π›½π‘˜1||π‘˜3(𝑑)𝐁1(𝑑).(3.54) Since πœ€π‘˜3(𝑑) is positive for π‘‘βˆˆπΌ, we have π‘˜βˆ—2β€–β€–β€–β€–(𝑓(𝑑))=π‘‘πβˆ—ξ€·π‘‘βˆ—ξ€Έπ‘‘π‘‘βˆ—||||π‘‘βˆ—=𝑓(𝑑)βˆ’π‘˜βˆ—1(𝑓(𝑑))π“βˆ—β€–β€–β€–β€–=ξ„Άξ„΅ξ„΅ξ„΅βŽ·(𝑓(𝑑))|||||βˆ’π›½2π‘˜22ξ€·π‘˜3ξ€Έ(𝑑)2||1+π›½π‘˜1||+ξ€·||1+π›½π‘˜1||π‘˜ξ€Έξ€·3(𝑑)2|||||=ξ”ξ€·π‘˜3ξ€Έ(𝑑)2=πœ€π‘˜3(𝑑)>0.(3.55) Thus, we can put 𝐁1βˆ—1(𝑓(𝑑))=π‘˜βˆ—2(𝑓(𝑑))π‘‘πβˆ—ξ€·π‘‘βˆ—ξ€Έπ‘‘π‘‘βˆ—||||π‘‘βˆ—=𝑓(𝑑)βˆ’π‘˜βˆ—1(𝑓(𝑑))π“βˆ—ξƒͺ(𝑓(𝑑))=βˆ’π›½π‘˜2||1+π›½π‘˜1||𝐓(𝑑)βˆ’||1+π›½π‘˜1||𝐁1(𝑑),π‘‘βˆˆπΌ.(3.56) By differentiation of the above equation with respect to 𝑑, we get π‘“ξ…ž(𝑑)𝑑𝐁1βˆ—ξ€·π‘‘βˆ—ξ€Έπ‘‘π‘‘βˆ—||||π‘‘βˆ—=𝑓(𝑑)=π‘˜2||1+π›½π‘˜1||𝐍(𝑑)βˆ’π‘˜3(𝑑)||1+π›½π‘˜1||𝐁2(𝑑).(3.57) Since π‘“ξ…žβˆš(𝑑)=|1+π›½π‘˜1| and π‘˜βˆ—2(𝑓(𝑑))πβˆ—(𝑓(𝑑))=π‘˜3(𝑑)𝐁2(𝑑), we have 𝑑𝐁1βˆ—ξ€·π‘‘βˆ—ξ€Έπ‘‘π‘‘βˆ—||||π‘‘βˆ—=𝑓(𝑑)+π‘˜βˆ—2(𝑓(𝑑))πβˆ—π‘˜(𝑓(𝑑))=2||1+π›½π‘˜1||𝐍(𝑑).(3.58) Thus, we obtain 𝐁2βˆ—(𝑓(𝑑))=𝛿𝐍(𝑑) for π‘‘βˆˆπΌ, where 𝛿=βˆ“1. We must determine whether 𝛿 is βˆ’1 or +1 under the condition that the frame field {π“βˆ—(𝑑),πβˆ—(𝑑),πβˆ—1(𝑑),πβˆ—2(𝑑)} is of positive orientation.
We have, by det[𝐓(𝑑),𝐍(𝑑),𝐁1(𝑑),𝐁2(𝑑)]=1 for π‘‘βˆˆπΌ,𝐓detβˆ—(𝑑),πβˆ—(𝑑),πβˆ—1(𝑑),πβˆ—2ξ€»βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£ξ”(𝑑)=det||1+π›½π‘˜1||𝐓(𝑑)+π›½π‘˜2||1+π›½π‘˜1||𝐁1(𝑑),πœ€π2(𝑑),βˆ’π›½π‘˜2||1+π›½π‘˜1||𝐓(𝑑)βˆ’||1+π›½π‘˜1||𝐁1⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦ξƒ©ξ€·||(𝑑),𝛿𝐍(𝑑)=πœ€π›Ώ1+π›½π‘˜1||ξ€Έβˆ’π›½2π‘˜22||1+π›½π‘˜1||ξƒͺ=πœ€π›Ώ(3.59) and det[π“βˆ—(𝑑),πβˆ—(𝑑),πβˆ—1(𝑑),πβˆ—2(𝑑)]=1 for any π‘‘βˆˆπΌ. Therefore, we get πœ€=𝛿. Thus, we get πβˆ—2π‘˜(𝑓(𝑑))=πœ€π(𝑑),βˆ—3ξ„”(𝑓(𝑑))=π‘‘πβˆ—1ξ€·π‘‘βˆ—ξ€Έπ‘‘π‘‘βˆ—||||π‘‘βˆ—=𝑓(𝑑),πβˆ—2ξ„•π‘˜(𝑓(𝑑))=πœ€2||1+π›½π‘˜1||,π‘‘βˆˆπΌ.(3.60) By the above facts, πΆβˆ— is a special Frenet curve in 𝐸41 and the first normal line at each point of 𝐢 is the third normal line of πΆβˆ— at corresponding each point under the bijection πœ™βˆΆπœ(𝑑)β†’πœ™(𝐜(𝑑))=πœβˆ—(𝑓(𝑑))βˆˆπΆβˆ—.
Thus, the proof is completed.

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

Theorem 3.6. Let 𝐢 be a timelike special curve defined by βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π›½ξ€œπ›½ξ€œπ›½ξ€œπ›½ξ€œβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦𝐜(𝑠)=𝑓(𝑠)cosh𝑠𝑑𝑠𝑓(𝑠)sinh𝑠𝑑𝑠𝑓(𝑠)𝑔(𝑠)𝑑𝑠𝑓(𝑠)β„Ž(𝑠)𝑑𝑠,π‘ βˆˆπ‘ˆβŠ‚β„.(3.61) Here, 𝛽 is a nonzero constant number, π‘”βˆΆπ‘ˆβ†’β„ and β„ŽβˆΆπ‘ˆβ†’β„ are any smooth functions, and the positive-valued smooth function π‘“βˆΆπ‘ˆβ†’β„ is given by 𝑓=1βˆ’π‘”2(𝑠)βˆ’β„Ž2ξ€Έ(𝑠)βˆ’3/2ξ‚€1βˆ’π‘”2(𝑠)βˆ’β„Ž2(𝑠)+̇𝑔2Μ‡β„Ž(𝑠)+2ξ€·Μ‡ξ€Έ(𝑠)βˆ’Μ‡π‘”(𝑠)β„Ž(𝑠)βˆ’π‘”(𝑠)β„Ž(𝑠)2ξ‚βˆ’5/2Γ—βŽ‘βŽ’βŽ’βŽ’βŽ£βˆ’ξ‚€1βˆ’π‘”2(𝑠)βˆ’β„Ž2(𝑠)+̇𝑔2Μ‡β„Ž(𝑠)+2ξ€·Μ‡ξ€Έ(𝑠)βˆ’Μ‡π‘”(𝑠)β„Ž(𝑠)βˆ’π‘”(𝑠)β„Ž(𝑠)23+ξ€·1βˆ’π‘”2(𝑠)βˆ’β„Ž2ξ€Έ(𝑠)3βŽ›βŽœβŽœβŽœβŽβˆ’(𝑔(𝑠)βˆ’Μˆπ‘”(𝑠))2βˆ’ξ€·Μˆξ€Έβ„Ž(𝑠)βˆ’β„Ž(𝑠)2βˆ’Μ‡Μ‡ξ€·ξ€·π‘”(𝑠)β„Ž(𝑠)βˆ’gξ€Έβˆ’ξ€·ΜˆΜ‡(𝑠)β„Ž(𝑠)̇𝑔(𝑠)β„Ž(𝑠)βˆ’Μˆπ‘”(𝑠)β„Ž(𝑠)ξ€Έξ€Έ2+ξ€·Μˆξ€Έπ‘”(𝑠)β„Ž(𝑠)βˆ’Μˆπ‘”(𝑠)β„Ž(𝑠)2⎞⎟⎟⎟⎠⎀βŽ₯βŽ₯βŽ₯⎦,(3.62) for π‘ βˆˆπ‘ˆ. Then, the curvature functions π‘˜1 and π‘˜2 of 𝐢 satisfy π‘˜1ξ€·π‘˜=βˆ’π›½21βˆ’π‘˜22ξ€Έ(3.63) at each point 𝐜(𝑠) of 𝐢.

Proof. Let 𝐢 be a timelike special curve defined by βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π›½ξ€œπ›½ξ€œπ›½ξ€œπ›½ξ€œβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦𝐜(𝑠)=𝑓(𝑠)cosh𝑠𝑑𝑠𝑓(𝑠)sinh𝑠𝑑𝑠𝑓(𝑠)𝑔(𝑠)𝑑𝑠𝑓(𝑠)β„Ž(𝑠)𝑑𝑠,π‘ βˆˆπ‘ˆβŠ‚β„,(3.64) where 𝛽 is a nonzero constant number 𝑔 and β„Ž are any smooth functions. 𝑓 is a positive-valued smooth function. Thus, we obtain Μ‡βŽ‘βŽ’βŽ’βŽ’βŽ£βŽ€βŽ₯βŽ₯βŽ₯⎦𝐜(𝑠)=𝛽𝑓(𝑠)cosh𝑠𝛽𝑓(𝑠)sinh𝑠𝛽𝑓(𝑠)𝑔(𝑠)𝛽𝑓(𝑠)β„Ž(𝑠),π‘ βˆˆπ‘ˆβŠ‚β„,(3.65) where the subscript prime (β‹…) denotes the differentiation with respect to 𝑠.
The arc-length parameter 𝑑 of 𝐢 is given byξ€œπ‘‘=πœ“(𝑠)=𝑠𝑠0β€–Μ‡πœ(𝑠)‖𝑑𝑠,(3.66) where β€–Μ‡βˆšπœ(𝑠)β€–=𝛽𝑓(𝑠)βˆ’1+𝑔2(𝑠)+β„Ž2(𝑠).
If πœ‘ denotes the inverse function of πœ“βˆΆπ‘ˆβ†’πΌβŠ‚β„, then 𝑠=πœ‘(𝑑) and we getπœ‘ξ…žβ€–β€–β€–(𝑑)=π‘‘πœ(𝑠)|||𝑑𝑠𝑠=πœ‘(𝑑)β€–β€–β€–βˆ’1,π‘‘βˆˆπΌ,(3.67) where the prime (ξ…ž) denotes the differentiation with respect to 𝑑.
The unit tangent vector 𝐓(𝑑) of the curve 𝐢 at the each point 𝐜(πœ‘(𝑑)) is given by𝐓(𝑑)=βˆ’1+𝑔2(πœ‘(𝑑))+β„Ž2ξ€Έ(πœ‘(𝑑))βˆ’1/2βŽ‘βŽ’βŽ’βŽ’βŽ£β„ŽβŽ€βŽ₯βŽ₯βŽ₯⎦cosh(πœ‘(𝑑))sinh(πœ‘(𝑑))𝑔(πœ‘(𝑑))(πœ‘(𝑑)),π‘‘βˆˆπΌ.(3.68) Some simplifying assumptions are made for the sake of brevity as follows: sinh∢=sinh(πœ‘(𝑑)),cosh∢=cosh(πœ‘(𝑑)),π‘“βˆΆ=𝑓(πœ‘(𝑑)),π‘”βˆΆ=𝑔(πœ‘(𝑑)),β„ŽβˆΆ=β„Ž(πœ‘(𝑑)),Μ‡π‘”βˆΆ=̇𝑔(πœ‘(𝑑))=𝑑𝑔(𝑠)|||𝑑𝑠𝑠=πœ‘(𝑑),Μ‡Μ‡β„ŽβˆΆ=β„Ž(πœ‘(𝑑))=π‘‘β„Ž(𝑠)|||𝑑𝑠𝑠=πœ‘(𝑑),π‘‘Μˆπ‘”βˆΆ=Μˆπ‘”(πœ‘(𝑑))=2𝑔(𝑠)𝑑𝑠2||||𝑠=πœ‘(𝑑),ΜˆΜˆπ‘‘β„ŽβˆΆ=β„Ž(πœ‘(𝑑))=2β„Ž(𝑠)𝑑𝑠2||||𝑠=πœ‘(𝑑),πœ‘ξ…žβˆΆ=πœ‘ξ…ž(𝑑)=π‘‘πœ‘|||𝑑𝑑𝑑,𝐴∢=1βˆ’π‘”2βˆ’β„Ž2Μ‡,𝐡∢=βˆ’π‘”Μ‡π‘”βˆ’β„Žβ„Ž,𝐢∢=βˆ’Μ‡π‘”2βˆ’Μ‡β„Ž2,ΜˆΜ‡β„ŽΜˆπ·βˆΆ=βˆ’π‘”Μˆπ‘”βˆ’β„Žβ„Ž,𝐸∢=βˆ’Μ‡π‘”Μˆπ‘”βˆ’β„Ž,𝐹∢=Μˆπ‘”2+Μˆβ„Ž2.(3.69) Thus, we get ̇̇̇𝐴=2𝐡,𝐡=𝐢+𝐷,𝐢=2𝐸,πœ‘ξ…ž=π›½βˆ’1π‘“βˆ’1π΄βˆ’1/2.(3.70) So, we rewrite (3.68) as π“βˆΆ=𝐓(𝑑)=π΄βˆ’1/2βŽ‘βŽ’βŽ’βŽ’βŽ£π‘”β„ŽβŽ€βŽ₯βŽ₯βŽ₯⎦.coshsinh(3.71) Differentiating the last equation with respect to 𝑑, we find π“ξ…ž=πœ‘ξ…žβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’12π΄βˆ’3/2̇𝐴cosh+π΄βˆ’1/2sinhβˆ’12π΄βˆ’3/2̇𝐴sinh+π΄βˆ’1/2coshβˆ’12π΄βˆ’3/2̇𝐴𝑔+π΄βˆ’1/2βˆ’1̇𝑔2π΄βˆ’3/2Μ‡β€ŒAβ„Ž+π΄βˆ’1/2Μ‡β„ŽβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,(3.72) that is, π“ξ…ž=βˆ’πœ‘ξ…žπ΄βˆ’1/2βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π΄βˆ’1𝐴𝐡coshβˆ’sinhβˆ’1𝐴𝐡sinhβˆ’coshβˆ’1π΄π΅π‘”βˆ’Μ‡π‘”βˆ’1Μ‡β„ŽβŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦.π΅β„Žβˆ’(3.73) From the last equation, we find π‘˜1∢=π‘˜1‖‖𝐓(𝑑)=ξ…žβ€–β€–(𝑑)=πœ‘ξ…žπ΄βˆ’1ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έ1/2.(3.74) By the fact that 𝐍(𝑑)=(π‘˜1(𝑑))βˆ’1π“ξ…ž(𝑑), we get 𝐍∢=𝐍(𝑑)=βˆ’π΄1/2ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έβˆ’1/2βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π΄βˆ’1𝐴𝐡coshβˆ’sinhβˆ’1𝐴𝐡sinhβˆ’coshβˆ’1π΄π΅π‘”βˆ’Μ‡π‘”βˆ’1Μ‡β„ŽβŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦.π΅β„Žβˆ’(3.75) In order to get second curvature function π‘˜2, we need to calculate π‘˜2(𝑑)=β€–πξ…ž(𝑑)βˆ’π‘˜1(𝑑)𝐓(𝑑)β€–. After a long process of calculations and using abbreviations, we obtain πξ…žβˆ’π‘˜1𝐓=πœ‘ξ…žπ΄βˆ’3/2ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έβˆ’3/2βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£Μ‡Μˆβ„ŽβŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦,(𝑃+𝑄)coshβˆ’π‘…sinh(𝑃+𝑄)sinhβˆ’π‘…coshπ‘ƒπ‘”βˆ’π‘…Μ‡π‘”+π‘„Μˆπ‘”π‘ƒβ„Žβˆ’π‘…β„Ž+𝑄(3.76) where 𝑃=π΄βˆ’π΄πΆ+𝐡2𝐡2ξ€Έβˆ’ξ€·βˆ’π΄πΆβˆ’π΄π·π΄βˆ’π΄πΆ+𝐡2ξ€Έ2+𝐴𝐡(π΅βˆ’π΄πΈ+𝐡𝐷),𝑄=𝐴2ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έ,𝑅=𝐴2(π΅βˆ’π΄πΈ+𝐡𝐷).(3.77) If we simplify 𝑃, then we have 𝑃=𝐴2(πΆβˆ’π΅πΈβˆ’π·+πΆπ·βˆ’1).(3.78) Therefore, we rewrite (3.76) and (3.77) as πξ…žβˆ’π‘˜1𝐓=πœ‘ξ…žπ΄βˆ’1/2ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έβˆ’3/2βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£ξ‚€ξ‚ξ‚π‘„ξ‚ξ‚ξ‚€ξ‚ξ‚π‘„ξ‚ξ‚ξ‚ξ‚ξ‚ξ‚ξ‚π‘…Μ‡ξ‚π‘„Μˆβ„ŽβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝑃+coshβˆ’π‘…sinh𝑃+sinhβˆ’π‘…coshπ‘ƒπ‘”βˆ’π‘…Μ‡π‘”+π‘„Μˆπ‘”π‘ƒβ„Žβˆ’β„Ž+(3.79) where 𝑃=πΆβˆ’π·+πΆπ·βˆ’π΅πΈβˆ’1,𝑄=π΄βˆ’π΄πΆ+𝐡2,𝑅=π΅βˆ’π΄πΈ+𝐡𝐷.(3.80) Consequently, from (3.79) and (3.80), we have β€–β€–πξ…žβˆ’π‘˜1𝐓‖‖2=ξ€·πœ‘ξ…žξ€Έ2π΄ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έβˆ’3βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’ξ‚€ξ‚ξ‚π‘„ξ‚π‘ƒ+2+𝑅2+𝑃2𝑔2+β„Ž2ξ€Έ+𝑅2̇𝑔2+Μ‡β„Ž2ξ€Έ+𝑄2ξ€·Μˆπ‘”2+Μˆβ„Ž2ξ€Έξ‚π‘ƒξ‚π‘…ξ€·Μ‡β„Žξ€Έξ‚π‘…ξ‚π‘„ξ€·Μ‡β„ŽΜˆβ„Žξ€Έξ‚π‘ƒξ‚π‘„ξ€·Μˆβ„Žξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦.βˆ’2𝑔̇𝑔+β„Žβˆ’2Μ‡π‘”Μˆπ‘”++2π‘”Μˆπ‘”+β„Ž(3.81) Substituting the abbreviations into the last equation, we have β€–β€–πξ…žβˆ’π‘˜1𝐓‖‖2=ξ€·πœ‘ξ…žξ€Έ2π΄ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έβˆ’3Γ—ξ‚ƒβˆ’ξ‚π‘ƒ2ξ‚π‘ƒξ‚ξ‚π‘„π΄βˆ’2π‘„βˆ’2+𝑅2βˆ’ξ‚π‘…2𝑄𝐢+2𝑃𝑅𝑃.𝐹+2𝑅𝐡+2π‘„πΈβˆ’2𝑄𝐷(3.82) After substituting (3.80) into the last equation and simplifying it, we get π‘˜22=β€–β€–πξ…žβˆ’π‘˜1𝐓‖‖2=ξ€·πœ‘ξ…žξ€Έ2π΄ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έβˆ’2ξ€Ίξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έ(1βˆ’πΉ)+(πΆβˆ’1)(1+𝐷)2βˆ’2𝐡𝐸(1+𝐷)+𝐴𝐸2ξ€».(3.83) Moreover, from (3.74) it is seen that π‘˜21=ξ€·πœ‘ξ…žξ€Έ2π΄βˆ’2ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έ.(3.84) The last two equations show us that π‘˜22βˆ’π‘˜21=ξ€·πœ‘ξ…žξ€Έ2π΄βˆ’2ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έβˆ’2Γ—ξ‚ƒβˆ’ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έ3+𝐴3ξ€·ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έ(1βˆ’πΉ)+(πΆβˆ’1)(1+𝐷)2βˆ’2𝐡𝐸(1+𝐷)+𝐴𝐸2ξ€Έξ‚„.(3.85) By the fact πœ‘ξ…ž=π›½βˆ’1π‘“βˆ’1π΄βˆ’1/2, we obtain π‘˜22βˆ’π‘˜21=π›½βˆ’2π‘“βˆ’2π΄βˆ’3ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έβˆ’2Γ—ξ‚ƒξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έ3+𝐴3ξ€·ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έ(1βˆ’πΉ)+(πΆβˆ’1)(1+𝐷)2βˆ’2𝐡𝐸(1+𝐷)+𝐴𝐸2,ξ€Έξ€»(3.86)π‘˜1=π›½βˆ’1π‘“βˆ’1π΄βˆ’3/2ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έ1/2.(3.87) According to our assumption 𝑓=1βˆ’π‘”2βˆ’β„Ž2ξ€Έβˆ’3/2ξ‚€1βˆ’π‘”2βˆ’β„Ž2+̇𝑔2+Μ‡β„Ž2βˆ’ξ€·Μ‡β„Žξ€ΈΜ‡π‘”β„Žβˆ’π‘”2ξ‚βˆ’5/2Γ—ξ‚Έβˆ’ξ‚€1βˆ’π‘”2βˆ’β„Ž2+̇𝑔2+Μ‡β„Ž2βˆ’ξ€·Μ‡β„Žξ€ΈΜ‡π‘”β„Žβˆ’π‘”23+ξ€·1βˆ’π‘”2βˆ’β„Ž2ξ€Έ3ξ‚€βˆ’(π‘”βˆ’Μˆπ‘”)2βˆ’ξ€·Μˆβ„Žξ€Έβ„Žβˆ’2βˆ’π‘”Μ‡ξ€Έβˆ’ξ€·ΜˆΜ‡β„Žξ€·ξ€·β„Žβˆ’Μ‡π‘”β„ŽΜ‡π‘”β„Žβˆ’Μˆπ‘”ξ€Έξ€Έ2+ξ€·π‘”Μˆξ€Έβ„Žβˆ’Μˆπ‘”β„Ž2,(3.88) we obtain 𝑓=π΄βˆ’3/2ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έβˆ’5/2βŽ‘βŽ’βŽ’βŽ’βŽ£ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έ3+𝐴3ξƒ©ξ€·π΄βˆ’π΄πΆ+𝐡2ξ€Έ(1βˆ’πΉ)+(πΆβˆ’1)(1+𝐷)2βˆ’2𝐡𝐸(1+𝐷)+𝐴𝐸2ξƒͺ⎀βŽ₯βŽ₯βŽ₯⎦.(3.89) Substituting the above equation into (3.86) and (3.87), we obtain π‘˜1ξ€·π‘˜=βˆ’π›½21βˆ’π‘˜22ξ€Έ.(3.90) The proof is completed.


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