BIHARMONIC CURVES IN MINKOWSKI 3-SPACE . PART II

This is a supplement to our previous research note [3]. In [3], we gave a characterization of biharmonic curves in Minkowski 3-space. More precisely, we pointed out that every biharmonic curves with nonnull principal normal in Minkowski 3-space is a helix, whose curvature κ and torsion τ satisfy κ2 = τ2. In the classification of biharmonic curves in Minkowski 3-space due to Chen and Ishikawa [1], there exist biharmonic spacelike curves with null principal normal. In this supplement, we give a characterization of biharmonic curves with null principal normal.


Introduction
This is a supplement to our previous research note [3].In [3], we gave a characterization of biharmonic curves in Minkowski 3-space.More precisely, we pointed out that every biharmonic curves with nonnull principal normal in Minkowski 3-space is a helix, whose curvature κ and torsion τ satisfy κ 2 = τ 2 .In the classification of biharmonic curves in Minkowski 3-space due to Chen and Ishikawa [1], there exist biharmonic spacelike curves with null principal normal.In this supplement, we give a characterization of biharmonic curves with null principal normal.

Preliminaries
Let E 3  1 be the Minkowski 3-space with natural Lorentz metric •, • = −dx 2 + dy 2 + dz 2 .Let γ = γ(s) be a spacelike curve parametrized by the arclength parameter; that is, γ satisfies γ ,γ = 1.A spacelike curve γ is said to be a Frenet curve if its acceleration vector field γ satisfies the condition γ ,γ = 0. Every spacelike Frenet curve admits an orthonormal frame field along it (see [3]).Since biharmonicity for spacelike Frenet curves is studied in [3], hereafter we restrict our attention to spacelike curves with null acceleration vector field.Note that spacelike curves with zero acceleration vector field are lines.There are no timelike curves with null acceleration vector field.Lemma 2.1.Let γ(s) be a spacelike curve parametrized by arclength such that γ ,γ = 0. Then there exists a matrix-valued function F(s) = (f 1 (s),f 2 (s),f 3 (s)), which satisfies the following ordinary differential equation: Here ∇ is the Levi-Civita connection of E 3 1 .Conversely, let F(s) = (f 1 (s),f 2 (s),f 3 (s)) be a solution to (2.1).Then there exists a spacelike curve γ(s) with arclength parameter s such that Proof.By the assumption, f 1 = γ is a null vector field.We set f 2 = f 1 .Since f 1 = γ is a unit spacelike vector field, there exists a unique null vector field f 3 along γ such that (cf.[2]) Hence f 2 = bf 2 .By similar computations, we get Conversely, let F be a solution to (2.1).Then F satisfies the following conditions (cf.[2, Section 2]): Integrating f 1 (s) by s, we obtain a spacelike curve γ(s) with null acceleration, since We call the matrix-valued function F, the null frame of γ.We call f 1 , f 2 , and f 3 , the tangent vector field, principal normal vector field, and binormal vector field of γ, respectively.We call the function k the curvature function of γ.Note that both principal normal and binormal are null.
Example 2.2.Let us consider γ with k = 0. Since f 2 = 0, we have where the constant vectors n and u satisfy the relation Thus we obtain where v is a constant vector.Hence γ is congruent to bs 2 ,bs 2 ,s , b = 0. (2.10) Next, assume that k is a nonzero constant, then γ is given by where the constant vectors n and u satisfy (2.8).Thus γ is congruent to the curve as 3 + bs 2 ,as 3 + bs 2 ,s , a = 0. (2.14)

Biharmonic curves
We start this section with recalling the notion of biharmonicity.
Let γ be a spacelike curve in E 3 1 parametrized by arclength defined on an open interval I.We denote by γ * TE 3  1 the vector bundle over I obtained by pulling back the tangent bundle TE 3  1 : The Laplace operator Δ acting on the space Γ(γ * TE 3 1 ) of all smooth vector fields along γ is given by A spacelike curve γ is said to be biharmonic if ΔH = 0, where H is the mean curvature vector field of γ.
Chen and Ishikawa obtained the following result.
Theorem 3.1 [1].Let γ(s) be a spacelike curve parametrized by arclength with null acceleration vector field.Then γ is biharmonic if and only if γ is congruent to Now we give a geometric characterization of biharmonic spacelike curve with null principal normal.Let γ(s) be a spacelike curve parametrized by arclength with null acceleration vector field.Then the mean curvature vector field H is given by (3.4) Thus we obtain Hence γ is biharmonic if and only if k + k 2 = 0. Hence the curvature function k is given by k = 0 or 1/k(s) = s + c, where c is a constant.
Proposition 3.2.A spacelike curve γ(s) parametrized by arclength parameter s with null principal normal vector field is biharmonic if and only if its curvature function is given by k = 0 or 1/k = s + c for some constant c.Hence such curves are congruent to the curve (3.3).The former case (k = 0) corresponds to the case a = 0 (2.10) and the latter case

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