inextensible flows of timelike curves with sabban frame in s2

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$on fractional moments of twists of l-functions attached to cusp forms$

6 V. Asil, T. Körpinar, S. Ba³⟨, ⟩ = −dx 2 1 + dx 2 2 + dx 2 3,where (x 1 , x 2 , x 3 ) is a rectangular coordinate system of E 3 1 . Recall that, thenorm of an arbitrary vector a ∈ E 3 1 is given by ∥a∥ = √ ⟨a, a⟩. γ is called aunit speed curve if velocity vector v of γ satises ∥a∥ = 1.Denote by {T, N, B} the moving Frenet-Serret frame along the timelikecurve γ in the space E 3 1 . For an arbitrary timelike curve γ with rst and secondcurvature, κ and τ in the space E 3 1 , the following Frenet-Serret formulaeare givenwhereT ′ = κN,N ′ = κT + τB,B ′ = −τN,⟨T, T⟩ = −1, ⟨N, N⟩ = ⟨B, B⟩ = 1,⟨T, N⟩ = ⟨T, B⟩ = ⟨N, B⟩ = 0.Here, the curvature functions are dened by κ = κ(s) = ∥T ′ (s)∥ and τ(s) =− ⟨N, B ′ ⟩.Torsion of the timelike curve γ is given by the aid of the mixed productτ = [γ′ , γ ′′ , γ ′′′ ]κ 2 .Now we give a new frame dierent from the Frenet frame. Let α : I → S 2 1be an unit speed spherical timelike curve. We denote σ as the arc-lengthparameter of α. Let us denote t (σ) = α ′ (σ), and we call t (σ) a unit tangentvector of α. We now set a vector s (σ) = α (σ) × t (σ) along α. This frameis called the Sabban frame of α on S 2 1 . Then we have the following sphericalFrenet-Serret formulae of α [3]:α ′ = t,t ′ = α + κ g s,s ′ = κ g t,where κ g is the geodesic curvature of the timelike curve α on the S 2 1 , andg (t, t) = −1, g (α, α) = 1, g (s, s) = 1,g (t, α) = g (t, s) = g (α, s) = 0.
Inextensible ows of timelike curves... 73. Inextensible ows of timelike curves according tothe Sabban frame in S 2 1Let α (u, t) is a one-parameter family of smooth timelike curves in S 2 1 .The arc length of α is given byσ(u) = u ∂α0 ∣ ∂u ∣ du,whereThe operator ∂∂σ∣⟨ ∂α∣∣∣ ∂α∣ ∂u ∣ = ∂u , ∂α ⟩∣ ∣∣∣1/2.∂uis given in terms of u by∂∂σ = 1 ∂ν ∂u ,where v =∂α∣ ∂u ∣ , and the arc length parameter is dσ = vdu.Any ow of α can be represented asLet the arc length variation be∂α∂t = fS 1 α + f S 2 t + f S 3 s. (3.1)σ(u, t) = u 0vdu.In the S 2 1 , the requirement that the curve not be subject to any elongationor compression can be expressed by the conditionfor all u ∈ [0, l] .∂∂t σ(u, t) = u ∂v0 du = 0 (3.2)∂tDefinition 3.1. The ow ∂α∂t in S2 1 is said to be inextensible if∂∂t∂α∣ ∂u ∣ = 0.
Page 1: ’iauliai Math. Semin.,7 (15), 201
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inextensible flows of timelike curves with sabban frame in s2

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