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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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