Harmonic maps and biharmonic maps on principal bundles and warped products. (English) Zbl 1414.58012
The paper under review studies biharmonicity of projections of principal G-bundles and the projection of a warped product onto its first factor. The main results include: (1) If the projection of a compact principal \(G\)-bundle over a compact based manifold with non-positive Ricci curvature is biharmonic, then it is harmonic. (2) If the projection of a non-compact complete \(G\)-bundle over a based manifold with non-positive Ricci curvature is a biharmonic map with finite energy and finite bienergy, then it is harmonic. Compared to Jiang’s general result saying that a biharmonic map from a compact manifold into a manifold of non-positive sectional curvature is harmonic, the result (1) weakens the assumption on the curvature but adds the compactness condition on the target manifold. The paper also derive the bitension field of a Riemannian submersion defined by the projection of a warped product onto its first factor. For some further study of biharmonic Riemannian submersions defined by the projections of warped products see [M. A. Akyol and the reviewer, “Biharmonic Riemannian submersions”, Ann. Mate. Pura Appl. 198, No. 2, 559–570 (2019; doi:10.1007/s10231-018-0789-x)].
Reviewer: Ye-Lin Ou (Commerce)
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