Characterization Of Inextensible Flows Of Spacelike Curves With Sabban Frame In S

Physically, inextensible curve and surface flows give rise to motions in which no strain energy is induced. The swinging motion of a cord of fixed length, for example, or of a piece of paper carried by the wind, can be described by inextensible curve and surface flows. Such motions arise quite naturally in a wide range of physical applications. This study is organised as follows: Firstly, we study inextensible flows of spacelike curves on S 1 . Secondly, we obtain partial differential equations in terms of inextensible flows of spacelike curves according to Sabban frame on S 1 .

Physically, inextensible curve and surface flows give rise to motions in which no strain energy is induced.The swinging motion of a cord of fixed length, for example, or of a piece of paper carried by the wind, can be described by inextensible curve and surface flows.Such motions arise quite naturally in a wide range of physical applications.
This study is organised as follows: Firstly, we study inextensible flows of spacelike curves on S 2 1 .Secondly, we obtain partial differential equations in terms of inextensible flows of spacelike curves according to Sabban frame on S 2 1 .

Preliminaries
The Minkowski 3-space E 3 provided with the standard flat metric given by where (x 1 , x 2 , x 3 ) is a rectangular coordinate system of E 3 1 .Recall that, the norm of an arbitrary vector a ∈ E 3 1 is given by a = a, a .γ is called a unit speed curve if velocity vector v of γ satisfies a = 1.
Denote by {T, N, B} the moving Frenet-Serret frame along the spacelike curve γ in the space E 3 1 .For an arbitrary spacelike curve γ with first and second curvature,

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Mahmut Ergüt, Essin Turhan and Talat Körpınar κ and τ in the space E 3 1 , the following Frenet-Serret formulae is given where Here, curvature functions are defined by κ = κ(s) = T ′ (s) and τ (s Torsion of the spacelike curve γ is given by the aid of the mixed product Now we give a new frame different from Frenet frame.Let α : I −→ S 2 1 be unit speed spherical curve.We denote σ as the arc-length parameter of α .Let us denote t (σ) = α ′ (σ) , and we call t (σ) a unit tangent vector of α.We now set a vector s (σ) = α (σ) × t (σ) along α.This frame is called the Sabban frame of α on S 2 1 .Then we have the following spherical Frenet-Serret formulae of α : where κ g is the geodesic curvature of the spacelike curve α on the S 2 1 and

Inextensible Flows of Curves According to Sabban Frame in S 2 1
Let α (u, t) is a one parameter family of smooth spacelike curves in S 2 1 .The arclength of α is given by where Characterization Of Inextensible Flows

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The operator ∂ ∂σ is given in terms of u by where v = ∂α ∂u and the arclength parameter is dσ = vdu.
Any flow of α can be represented as Letting the arclength variation be In the S 2 1 the requirement that the curve not be subject to any elongation or compression can be expressed by the condition for all u ∈ [0, l] .
Definition 3.1.The flow ∂α ∂t in S 2 1 are said to be inextensible if s be a smooth flow of the spacelike curve α.The flow is inextensible if and only if Proof: Suppose that ∂α ∂t be a smooth flow of the spacelike curve α.
From (3.3), we obtain By the formula of the Sabban, we have Making necessary calculations from above equation, we have (3.5), which proves the lemma. 2 3 s be a smooth flow of the spacelike curve α.The flow is inextensible if and only if Proof: From (3.4), we have We now restrict ourselves to arc length parametrized curves.That is, v = 1 and the local coordinate u corresponds to the curve arc length σ.We require the following lemma.
Proof: Using definition of α, we have Using the Sabban equations, we have Characterization Of Inextensible Flows

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On the other hand, substituting (3.7) in (3.12), we get Since, we express : Then, a straight forward computation using above system gives where ψ = ∂α ∂t , s .Thus, we obtain the theorem.

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The following theorem states the conditions on the curvature and torsion for the flow to be inextensible.Theorem 3.5.Let ∂α ∂t is inextensible.Then, the following system of partial differential equations holds: Proof: Using (3.9), we have On the other hand, from Sabban frame we have Hence we see that Thus, we obtain the theorem.
s be a smooth flow of the spacelike curve α.Then, Proof: Similarly, we have On the other hand, a straightforward computation gives Combining these we obtain the corollary.2 In the light of Theorem 3.6, we express the following corollary without proof: Corollary 3.7.