Triharmonic Curves on 3-Dimensional Strict Walker Manifolds
Münevver Yıldırım Yılmaz *
Department of Mathematics, Firat University, 23119, Elaziğ, Türkiye.
Ayşe Dolu
Graduate School of Natural and Applied Sciences, Firat University, 23119, Elaziğ, Türkiye.
*Author to whom correspondence should be addressed.
Abstract
Biharmonic curves in three-dimensional strict Walker manifolds have already attracted some attention in the literature. However the corresponding theory of triharmonic curves has not yet been addressed. The main motivation of this paper is to fill this gap by studying triharmonic curves in three-dimensional strict Walker manifolds and present explicit characterizations of triharmonic curves and their geometric properties. In this paper, triharmonic curves in three-dimensional strict Walker manifolds endowed with a Lorentzian metric are investigated. After recalling the geometric structure of strict Walker manifolds and polyharmonic curves, the triharmonic curve equation is formulated in the Lorentzian setting. Also the necessary and sufficient conditions for a curve to be triharmonic in a three-dimensional strict Walker manifold are given and such curves are characterized in terms of their curvature and torsion functions. The obtained results extend the theory of harmonic and biharmonic curves to the higher-order variational framework of triharmonic maps within the Walker Lorentzian context.
Keywords: Triharmonic curve, strict walker manifold, lorentzian manifold, frenet frame, walker metric