Vector fields which are biharmonic maps
A biharmonic map is a map \(\phi: (M^m, g)\to (N^n, h)\) between Riemannian manifolds which is a critical point of the bienergy functional \(\frac{1}{2}\int_M\,|\tau(\phi)|^2\;dv_g\), where \(\tau(\phi)\) is the tension field of the map. For some recent study and progress on biharmonic maps and the related topics see a recent book [\textit{Y.-L. Ou} and \textit{B.-Y. Chen}, Biharmonic submanifolds and biharmonic maps in Riemannian geometry. Hackensack, NJ: World Scientific (2020; Zbl 1455.53002)] and the the vast references therein. The paper under review studies biharmonicity of vector fields on a manifold treated as maps \(X: (M, g)\to (TM, g_S)\) from the manifold into its tangent bundle provided with the Sasaki metric. The main results include an explicit biharmonic equation for such maps, and a theorem shows that a vector field on a compact manifold defines a biharmonic map into its tangent bundle if and only if the vector field is parallel. The paper can be viewed as a generalization of the work in [\textit{M. Markellos} and \textit{H. Urakawa}, Monatsh. Math. 178, No. 3, 389--404 (2015; Zbl 1328.58010)], where the authors defined biharmonic vector fields as critical points of the bienergy through variations of maps defined by vector fields (so a special class of variations), and proved that a biharmonic vector field on a compact manifold is parallel.
scientific article; zbMATH DE number 7493132
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