## Biharmonic Legendre curves in Sasakian space forms.(English)Zbl 1152.53049

Let $$f: M\rightarrow N$$ be a smooth map between two Riemannian manifolds. The bi-energy of $$f$$ is defined by $$E_2(f)= \frac{1}{2}\int_M\|\text{trace}\nabla f\|^2*1$$. We say that $$f$$ is a biharmonic map if it is a critical point of the bienergy. A normal contact metric manifolds is called a Sasaki manifold. If a manifold admits three almost contact structures $$(\varphi_{a},\xi_{a},\eta_{a})$$, satisfying $\varphi_c= \varphi_a\varphi_b- \eta_b\otimes\xi_a= -\varphi_b\varphi_a+ \eta_a\otimes\xi_b,$
$\xi_c= \varphi_a\xi_b= -\varphi_b\xi_a,\;\eta_c= \eta_a\circ\varphi_b= -\eta_b\circ\varphi_a$ for any even permutation $$\{a,b,c\}$$ of $$\{1,2,3\}$$, then the manifold is said to have an almost contact 3-structure. If all structures are Sasakian then we call the manifold a 3-Sasakian.
In this paper biharmonic Legendre curves in a Sasakian space form are studied. Explicit formulas for some biharmonic Legendre curve in the 7-sphere are given. Some rigidity results in 7-dimensional 3-Sasakian manifolds was obtained, for example
Proposition 4.2. A biharmonic Legendre curve with respect to all three Sasakian structures on $$S^7$$, is a geodesic.

### MSC:

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