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Biharmonic timelike curves according to Bishop frame in Minkowski 4-space

Solouma EM, Wageeda MM

In the last decade there has been a growing interest in the theory of biharmonic maps which can be divided in two main research directions. On the one side, constructing the examples and classification results has become important from the differential geometric aspect. The other side is the analytic aspect from the point of view of partial differential equations, because biharmonic maps are solutions of a fourth order strongly elliptic semi linear PDE.Biharmonic curves γ : I ?⊂???→(N, h) of a Riemannian manifold are the solutions of the fourth order differential equation 3 R( , 3 ) 0, γ γ γ γ γ γ ′ ′ ∇ ′ − ′ ∇ ′ ′ = (1) where,∇ is the Levi-Civita connection on (N, h) and R is its curvature operator. As we shall detail in the next section, they arise from a variational problem and are a natural generalization of geodesics. In the last decade biharmonic curves have been extensively studied and classified in several spaces by analytical inspection of Equation 1 [1-15].Although much work has been done, the full understanding of biharmonic curves in a surface of the Euclidean threedimensional space is far from been achieved. As yet, we have a clear picture of biharmonic curves in a surface only in the case that the surface is invariant by the action of a one parameter group of isometries of the ambient space. For example, in the study by Caddeo et al. [4] it was proved that a biharmonic curve on a surface of revolution in the Euclidean space must be a parallel that is an orbit of the action of the group on the surface. This property was then generalized to invariant surfaces in a 3-dimensional manifold [15].In this paper, we study biharmonic non-lightlike curves according to Bishop frame in Minkowski 4-space 4 1 E . We give some characterizations for curvatures of a biharmonic non-lightlike curve in 4 1 E . This paper is organized as follows: Section 2 gives some basic concepts of the Frenet frame and Bishop frame of a curve in 4 1 E . Section 3 obtained some characterizations for curvature of these curves with respect to the principal curvature functions ( ) 1ks , ( ) ( ) 2 3 k s , k s according to Bishop frame