No abstract

A classical theorem by Lancret says that a curve in Euclidean 3-space is of constant slope if and only if its ratio of curvature and torsion is constant. In this paper we study Lancret type problems for curves in Sasakian 3-manifolds.

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.

We show that a non-Sasakian contact metric manifold with η-parallel torsion tensor and sectional curvatures of plane sections containing the Reeb vector field different from 1 at some point, is a (k, µ)-contact manifold. In particular for the standard contact metric structure of the tangent sphere bundle the torsion tensor is η-parallel if and only if M is of constant curvature, in which case its associated pseudo-Hermitian structure is CRintegrable. Next we show that if the metric of a non-Sasakian (k, µ)-contact manifold (M, g) is a gradient Ricci soliton, then (M, g) is locally flat in dimension 3, and locally isometric to E n+1 × S n (4) in higher dimensions.

By using the pseudo-Hermitian connection (or Tanaka-Webster connection) ∇, we construct the parametric equations of Legendre pseudo-Hermitian circles (whose ∇-geodesic curvature κ is constant and ∇-geodesic torsion τ is zero) in S 3 . In fact, it is realized as a Legendre curve satisfying the ∇-Jacobi equation for the ∇-geodesic vector field along it.2000 Mathematics subject classification: 53C25, 53C43, 54D10.