Biharmonic Reeb curves in Sasakian manifolds

TL;DR

This paper characterizes non-geodesic biharmonic curves in Sasakian manifolds, especially those tangent or normal to the Reeb vector field, revealing their structure as helixes with specific curvature-torsion relations.

Contribution

It provides a complete characterization of certain biharmonic curves in Sasakian manifolds, including explicit relations in three dimensions.

Findings

Biharmonic curves tangent or normal to the Reeb vector field are helixes.

In three dimensions, these curves satisfy specific curvature and torsion relations.

Explicit formulas for Jacobi operators in Sasakian manifolds facilitate this characterization.

Abstract

Sasakian manifolds provide explicit formulae of some Jacobi operators which describe the biharmonic equation of curves in Riemannian manifolds. In this paper we characterize non-geodesic biharmonic curves in Sasakian manifolds which are either tangent or normal to the Reeb vector field. In the three-dimensional case, we prove that such curves are some helixes whose geodesic curvature and geodesic torsion satisfy a given relation.