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Biharmonic curves into quadrics. (English) Zbl 1322.58010
Biharmonic curves are special cases of biharmonic maps, which are maps between Riemannian manifolds that are critical points of the bienergy functional. An arclength parametrized curve \(\gamma: I\subset \mathbb{R}\to (N^n, h)\) is biharmonic if and only if it solves the \(4\)-th order differential equation \(\nabla^3_{\gamma'}\gamma'-\text{R}^N(\gamma', \nabla_{\gamma'}\gamma')\gamma'=0\). Clearly, any geodesic is a biharmonic curve.
In recent years, a lot of work has been done in classifying biharmonic curves in certain model spaces, see, e.g., R. Caddeo et al. [Int. J. Math. 12, No. 8, 867–876 (2001; Zbl 1111.53302); Mediterr. J. Math. 3, No. 3–4, 449–465 (2006; Zbl 1116.58013); Rend. Mat. Appl., VII. Ser. 21, No. 1–4, 143–157 (2001; Zbl 1049.58020)]; I. Dimitrić [Bull. Inst. Math., Acad. Sin. 20, No. 1, 53–65 (1992; Zbl 0778.53046)]; D. Fetcu [Ann. Mat. Pura Appl. (4) 189, No. 4, 591–603 (2010; Zbl 1201.53070)].
The paper under review studies biharmonic curves into quadrics \(Q_1=\{(x,y,z)|\; \frac{x^2}{a^2}\pm \frac{y^2}{b^2}\pm \frac{z^2}{c^2}=1\}\) (with a center) and \(Q_2=\{(x,y,z)|\; \frac{x^2}{a^2}\pm \frac{y^2}{b^2}=2z\}\) (without a center). The main results show that a quadric with a center admits a proper biharmonic curve if and only if it is \(\{(x,y,z)|\; \frac{x^2}{a^2}+ \frac{y^2}{a^2}+ \frac{z^2}{c^2}=1\}\), and in this case, the biharmonic curve is the intersection of the quadric with the ellipsoid \(\{(x,y,z)|\; \frac{x^2}{a^4}+ \frac{y^2}{a^4}+ \frac{z^2}{c^4}=\frac{1}{ac}\}\); there exists no proper biharmonic curve in a quadric without a center.

MSC:
58E20 Harmonic maps, etc.
53C43 Differential geometric aspects of harmonic maps
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