Biharmonic Curves in a Strict Walker 3-Manifold

Mamadou Gningue,Ameth Ndiaye,Rénovat Nkunzimana

doi:10.1155/2022/3855033

Abstract

In this paper, we study the geometry of biharmonic curves in a strict Walker 3-manifold and we obtain explicit parametric equations for biharmonic curves and time-like biharmonic curves, respectively. We discuss the conditions for a speed curve to be a slant helix in a Walker manifold. We give an example of biharmonic curve for illustrating the main result.

Highlights

Eells and Sampson [4] defined harmonic and biharmonic map between Riemannian manifolds
Lee [10] studies biharmonic curves in 3-dimensional Lorentzian Heisenberg space (H3; g), and he shows that proper biharmonic space-like curve c in Lorentzian Heisenberg space (H3; g) is pseudohelix with some properties about their curvatures
When the target manifold is semi-Riemannian manifold, the bienergy and bitension field can be defined in the same way

Summary

Preliminaries

When the target manifold is semi-Riemannian manifold, the bienergy and bitension field can be defined in the same way. Walker has derived adapted coordinates to a parallel plan field. The metric of a three-dimensional Walker manifold (M, gεf) with coordinates (x, y, z) is expressed as gεf dx ∘ dz + εdy2 + f(x, y, z)dz. If (M, gεf) is a strict Walker manifolds, i.e., f(x, y, z) f(y, z), the associated Levi-Civita connection satisfies (10). Note that the existence of a null parallel vector field (i.e., f f(y, z)) simplifies the nonzero components of the Christoffel symbols and the curvature tensor of the metric gεf as follows: Γ123. E vector product of u and v in (M, gεf) with respect to the metric gεf is the vector denoted by u×fv in M defined by gεf􏼐u×fv, w􏼑 det(u, v, w),. Lemma 1. e Walker cross product in M has the following properties:

Biharmonic Curves in Walker 3-Manifold

Slant Helix in Walker 3-Manifold