The Euler-Lagrange Method for Biharmonic Curves

Abstract

In this article we consider the Euler-Lagrange method associated to a suitable bilagrangian to study biharmonic curves of a Riemannian manifold. We apply this method to characterize biharmonic curves of the threedimensional Lie group Sol. We also classify, using a geometric method, the biharmonic curves of the three-dimensional Cartan-Vranceanu manifolds.2000 Mathematics Subject Classification. 58E20.

Biharmonic Curves in Finsler Spaces

“…Since in the case when κ 1 = 0 (i.e., c is a geodesic), the curve is trivially biharmonic, we immediately get: A similar result is known to hold in Riemannian geometry (see [2], [6], [14], [18]). …”

“…In Riemannian spaces, biharmonic curves and, more generally, biharmonic maps, were examined quite in detail (see [6], [8], [12], [13], [14], [17], [18], [19]). As a first remark, geodesics of ∇ are always biharmonic (actually, they are minimum points for the bienergy), but the converse is generally not true.…”

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.…”

On the Geometry of Biharmonic Submanifolds in Sasakian Space Forms

“…There are several classification results for the proper-biharmonic submanifolds in Euclidean spheres and non-existence results for such submanifolds in space forms N c , c ≤ 0 ( [4], [5], [7], [8], [9], [10], [13]), while in spaces of non-constant sectional curvature only few results were obtained ( [1], [12], [18], [19], [25], [29]). …”