Abstract
In this paper, we firstly derived the equations for the curves of a Lorentz–Minkowski space L3 to be f-biharmonic. Then, using these equations, we classify such unit speed curves in L3.
Highlights
IntroductionBiharmonic isometric immersions are critical points of the bienergy functional (proposed by Eells and Lemaire in [1])
For isometric immersions φ: Mnr ⟶ Nnq+p from an n-dim(ne+nspio)n-dailmepnsesiuodnoa-lRipesme→uadnon-iRaniemmananniiafonldm→aMninrfolidntoNnq+apn, where τ(φ) tr∇dφ nH, with H be the mean curvature vector field of Mnr, is the tension field of φ vanishing of which means that φ is harmonic or Mnr is minimal
E first variation formula for the bienergy E2(φ) which is derived by Jiang in [4] shows that the Euler–Lagrange equation for E2(φ) is τ2(φ) ≔ trace∇φ∇φ − ∇φ∇τ(φ) − traceR(dφ, τ(φ))dφ 0, (2)
Summary
Biharmonic isometric immersions are critical points of the bienergy functional (proposed by Eells and Lemaire in [1]). Caddeo, Montaldo, and Oniciuc (cf [2]) showed nonexistence of nongeodesic biharmonic curves in a 3-dimensional hyperbolic space and proved that nongeodesic biharmonic curves in the unit 3-sphere are circles of geodesic curvature 1 or helices which are geodesics in the Clifford minimal torus. Ou in [8] derived equations for f-biharmonic curves in a generic manifold and completely classified f-biharmonic curves in 3-dimensional Euclidean space E3, where he proved that such curves in E3 are planar curves or general helices and gave some examples of nonbiharmonic f-biharmonic curves in E3. We will investigate unit speed f-biharmonic curves with a positive function f in Lorentz–Minkowski space L3 and obtain the following classification theorems. (iii) c is a helix curve and f (c1 + c2s)e− τds with τ being nonconstant where τ is the torsion of c, c1, and c2 are two constants
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