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Biharmonic curves in 3-dimensional Sasakian space forms. (English) Zbl 1141.53060

A smooth map between two Riemannian manifolds is said to be a biharmonic map if its bitension field vanishes identically. Let \(M\) be a 3-dimensional Sasakian space form of constant holomorphic sectional curvature \(H\). Non-geodesic biharmonic curves of \(M\) are called proper biharmonic curves. In this paper the authors prove that every proper biharmonic curve of \(M\) is a helix \(\gamma\) (with constant geodesic curvature \(k\) and geodesic torsion \(\tau\)). In particular, if \(H\neq 1\), \(\gamma\) is a helix which makes a constant angle \(\alpha\) with the Reeb vector field such that \(k^2 + \tau^2 = 1 + (H-1)\sin^2\alpha\). Moreover, they give explicitly parametric equations of proper biharmonic helices in the so-called Bianchi-Cartan-Vranceanu model spaces of Sasakian spaces forms.
Reviewer: D. Perrone (Lecce)

MSC:

53C43 Differential geometric aspects of harmonic maps
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C40 Global submanifolds
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