tailieunhanh - On biharmonic legendre curves in S-space forms

In the paper, we study biharmonic legendre curves in S−space forms. We find curvature characterizations of these special curves in 4 cases. The paper is organized as follows: In section 2, we give a brief introduction about S−space forms. In section 3, we give the main results of the study. | Turkish Journal of Mathematics Turk J Math (2014) 38: 454 – 461 ¨ ITAK ˙ c TUB ⃝ doi: Research Article On Biharmonic Legendre curves in S -space forms ¨ ¨ ∗, S ¨ Cihan OZG UR ¸ aban GUVENC ¸ Department of Mathematics, Balıkesir University, C ¸ a˘ gı¸s, Balıkesir, Turkey Received: • Accepted: • Published Online: • Printed: Abstract: We study biharmonic Legendre curves in S− space forms. We find curvature characterizations of these special curves in 4 cases. Key words: S− space form, Legendre curve, biharmonic curve, Frenet curve 1. Introduction Let (M, g) and (N, h) be 2 Riemannian manifolds and f : (M, g) → (N, h) a smooth map. The energy functional of f is defined by ∫ 1 2 |df | υg . E(f ) = 2 M If f is a critical point of the energy functional E(f ), then it is called harmonic [10]. f is called a biharmonic map if it is a critical point of the bienergy functional 1 E2 (f ) = 2 ∫ 2 |τ (f )| υg , M where τ (f ) is the first tension field of f , which is defined by τ (f ) = trace∇df. The Euler-Lagrange equation of bienergy functional E2 (f ) gives the biharmonic map equation [16] τ2 (f ) = −J f (τ (f )) = −∆τ (f ) − traceRN (df, τ (f ))df = 0, where J f is the Jacobi operator of f . It is trivial that any harmonic map is biharmonic. If the map is a nonharmonic biharmonic map, then we call it proper biharmonic. Biharmonic submanifolds have been studied by many geometers. For example, see [2], [3], [7], [8], [11], [12], [13], [14], [15], [18], [20], [21], [22], and the references therein. In a different setting, in [9], Chen defined a biharmonic submanifold M ⊂ En of the Euclidean space as its mean curvature vector field H satisfies ∆H = 0 , where ∆ is the Laplacian. In [12] and [14], Fetcu and Oniciuc studied biharmonic Legendre curves in Sasakian space forms. As a generalization of their studies, in the present paper, we study biharmonic Legendre curves .