On new characterization of inextensible flows of space-like curves in de Sitter space

Abstract Elastica and inextensible flows of curves play an important role in practical applications. In this paper, we construct a new characterization of inextensible flows by using elastica in space. The inextensible flow is completely determined for any space-like curve in de Sitter space S 1 3 $\mathbb{S}_{1}^{3}$. Finally, we give some characterizations for curvatures of a space-like curve in de Sitter space S 1 3 $\mathbb{S}_{1}^{3}$.


Introduction
In mathematics and physics, a de Sitter space is the analogue in Minkowski space, or spacetime, of a sphere in ordinary, Euclidean space. The n-dimensional de Sitter space, denoted dS n , is the Lorentzian manifold analogue of an n-sphere (with its canonical Riemannian metric). It is maximally symmetric, has constant positive curvature, and is simply connected for n at least 3. More recently, it has been considered as the setting for special relativity rather than using Minkowski space, since a group contraction reduces the isometry group of de Sitter space to the Poincaré group, allowing a unification of the spacetime translation subgroup and Lorentz transformation subgroup of the Poincaré group into a simple group rather than a semi-simple group. This alternate formulation of special relativity is called de Sitter relativity, [1][2][3][4][5][6].
The elastica caught the attention of many of the brightest minds in the history of mathematics, including Galileo, the Bernoullis, Euler, and others. It was present at the birth of many important fields, most notably the theory of elasticity, the calculus of variations, and the theory of elliptic integrals. The path traced by this curve illuminates a wide range of mathematical style, from the mechanics-based intuition of the early work, through a period of technical virtuosity in mathematical technique, to the present day where computational techniques dominate [16][17][18][19][20].
The flow of a curve or surface is said to be inextensible if, in the former case, the arc-length is preserved, and in the latter case, if the intrinsic curvature is preserved [7]. Physically, inextensible curve and surface flows are characterized by the absence of any strain energy induced from the motion. In [15], Kwon investigated inextensible flows of curves and developable surfaces in R 3 . Necessary and sufficient conditions for an inextensible curve flow were first expressed as a partial differential equation involving the curvature and torsion. Then, they derived the corresponding equations for the inextensible flow of a developable surface, and showed that it suffices to describe its evolution in terms of two inextensible curve flows, [15]. Flows of curves of a given curve are also widely studied, [8][9][10][11][12][13][14].
This study is organised as follows: Firstly, we construct a new method for inextensible flows of space-like curves in de Sitter space S 3 1 . Secondly, using the Frenet frame of the given curve, we present partial differential equations. Finally, we give some characterizations for curvatures of a curve in de Sitter space S 3 1 : 2 New geometry of space-like curves in S 3

-space
It is well-known that the Lorentzian space form with a positive curvature, more precisely [1], a positive sectional curvature is called de Sitter space S 3 1 . We define de Sitter 3-space by It is well-known that to each unit speed space-like curve W I ! S 3 1 one can associate a pseudo-orthonormal frame f ; T; N; Bg : Denote by T; N; B the space-like tangent vector, the space-like principal normal vector, and the time-like binomial vector, respectively. In this situation, the Frenet-Serret equations satisfied by the Frenet vectors T; N; B formally given by where ı. / D sign.N/, and Ä; are the curvature and the torsion of a curve respectively and given by .s/ D ı. / R 2 det. ; 0 ; 00 ; 000 /; with R.s/ ¤ 0: Let .u; w/ be a one parameter family of smooth space-like curves in S 3 1 : @ @w Finally [18], we obtain that W.v/ D gv; g D < r T W; T > : Theorem 2.2. Let @ @w be a smooth flow of : The flow is inextensible if and only if Now, assume that is arc-length parametrized curve. Then, we have where 1 ; 2 ; 3 are smooth functions of time and arc-length.
Proof. From the definition of inextensible flow, we have Using Eq. (3), we obtain Eq. (4). This completes the proof.
Now we give the characterization of evolution of first curvature as below: Theorem 2.4. Let be one parameter family curves in de Sitter space S 3 1 . If @ @w is inextensible flow of space-like in de Sitter space S 3 1 , then the evolution of Ä is given by where 1 ; 2 ; 3 are smooth functions of time and arc-length.
Proof. A differentiation in Eq. (4) and the Frenet formulas give us that Using the formula of the curvature, we write a relation r w r s T r s r w T D R. @ @s ; @ @w /T: We immediately arrive at R. @ @s ; @ @w /T D 2 R.T; N/TC 3 R.T; B/T: Another important fact is that the curvature operator R on de Sitter space S 3 1 has a simple expression, i.e., Then, From above equations, we get R. @ @s ; @ @w /T D 2 NC 3 B: Then, we can write r w r s T D r s r w T C R. @ @s ; @ @w /T: Thus it is easy to obtain that On the other hand, we have @Ä @w D @ @w g .r s T; N/ : Since, we express @Ä @w D g .r w r s T; N/ Cg .r s T;r w N/ : Moreover, by the definition of metric tensor, we have g .N;r w N/ D0: Then g .r s T;r w N/ D g . ;r w N/ D 1 Ä @ 1 @s : Combining these we have Thus, we obtain the theorem. This completes the proof.
From the above theorem, we have Theorem 2.5.
where 1 ; 2 ; 3 are smooth functions of time and arc-length.
Proof. Using Frenet equations, we have r w r s T D @ @w C @Ä @w NCÄr w N: Then, Therefore g .N;r w N/ D0: From above equation we obtain which completes the proof.
Theorem 2.6. Let be one parameter family curves in de Sitter space S 3 1 . If @ @w is inextensible flow of space-like in de Sitter space S 3 1 , then where 1 ; 2 ; 3 are smooth functions of time and arc-length.
Proof. Assume that @ @w be inextensible flow of : Under the assumption of space-like curve, we have Using the formula of the curvature, we write a relation r w r s N r s r w N D R. Arranging the last equations, we obtain R. @ @s ; @ @w /N D 2 T: Therefore, we can easily see that OEÄı. /OE 2 C @ 3 @s C @ @t B: By this way, we conclude g .B 1 ;r w B 1 / D0: Thus, we obtain the theorem. The proof of theorem is completed.
Now we give the characterization of evolution of second curvature as below: Theorem 2.7. Let be one parameter family curves in de Sitter space S 3 1 . If @ @t is inextensible flow of space-like in de Sitter space S 3 1 , then the evolution of is given by where 1 ; 2 ; 3 are smooth functions of time and arc-length.
Proof. It is obvious from Theorem 2.6. This completes the proof.
Since .t/ is an immersed curve, it has velocity vector V D vT and squared geodesic curvature Theorem 2.8 (Main Theorem).
where g D < r T W; T > : Proof. From Euler equations, we easily have W.Ä 2 C 1/ D 2 < r s r s W; r s T > C2 < R.W; T/T; r s T > C4g < r s T; r s T > : In what follows, W OE0; 1 ! M is a curve of length L: Now for fixed constant let For a variation w with variation field W, we compute This condition implies that where E D .r s / 3 T r s .ƒT/ C R.r s T; T/T;g D < r T W; T > and ƒ D 3Ä 2 4 2 : Thus, we can state the following.
where 1 ; 2 ; 3 are smooth functions of time and arc-length.
Proof. Firstly, we obtain Since, we immediately arrive at R. @ @s ; @ @t /T D 2 NC 3 B: Therefore, Now, we can obtain following equation in terms of flows. where A, B are functions only of time.
Projection of at xyz-plane: