Biharmonic curves in 3-dimensional Sasakian space forms

Abstract

We show that every proper biharmonic curve in a 3-dimensional Sasakian space form of constant holomorphic sectional curvature H is a helix (both of whose geodesic curvature and geodesic torsion are constants). In particular, if H = 1, then it is a slant helix, that is, a helix which makes constant angle α with the Reeb vector field with the property κ 2 + τ 2 = 1 + (H − 1) sin 2 α. Moreover, we construct parametric equations of proper biharmonic herices in Bianchi-Cartan-Vranceanu model spaces of a Sasakian space form.

The Euler-Lagrange Method for Biharmonic Curves

“…In particular, if m = 0 and = 0, (M, ds 2 ,m ) is the Heisenberg group H 3 endowed with a left invariant metric and the explicit solutions of the biharmonic curves were obtained in [5]; if = 2 and m = 0 a study of the biharmonic curves was given in [8]; the characterization we propose in this paper was also communicated in [6].…”

Explicit formulas for biharmonic submanifolds in Sasakian space forms

“…Further, the proper-biharmonic curves of ‫ޓ‬ n , n > 3, are, up to a totally geodesic embedding of ‫ޓ‬ 3 in ‫ޓ‬ n , those of ‫ޓ‬ 3 [Caddeo et al 2002]. Classification results for proper-biharmonic curves in 3-dimensional spaces of nonconstant sectional curvature were obtained in [Caddeo et al 2006;Cho et al 2007;Fetcu and Oniciuc 2007;Inoguchi 2004], and it turn out that, in the studied cases, they are helices.…”

Slant and Legendre curves in Bianchi-Cartan-Vranceanu geometry

“…They form a two parameters family (with parameters denoted as l and m) containing, among others, some remarkable 3-manifolds: R 3 , S 3 , S 2 × R, H 2 × R and the 3-dimensional Heisenberg group Nil 3 . Recently, several studies have been devoted to special submanifolds in these spaces: parallel surfaces [2], biharmonic curves [6], [12] and [13], constant angle surfaces [15], graphs of constant mean curvature [19], biharmonic surfaces [21], higher order parallel and totally umbilical surfaces [23].…”