## 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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### References:

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