Harmonic maps and biharmonic Riemannian submersions. (English) Zbl 1432.58012
Biharmonic maps are critical points of the functional
\[
E_2(\phi) = \int_M |\tau(\phi)|^2 \, v_g
\]
where \(\tau(\phi)\) is the tension field of \(\phi\), and generalise harmonic maps.
This article studies the biharmonicity of the projection \(\pi : (P,g) \to (M,h)\) where \((P,g)\) is an \({\mathbb S}^1\)-bundle over a compact Riemannian manifold \((M,h)\) and \(\pi\) is a Riemannian submersion.
First, it is proved that for a base manifold with non-positive Ricci curvature, when \(\pi\) is biharmonic, if the tension field of \(\pi\) is divergence-free then it must be parallel.
The cases of a weakly stable Einstein base manifold and of an irreducible compact Hermitian symmetric space are studied in more details and characterisations of the biharmonicity of the bundle projection are given.
This article closes with explicit examples of Kähler-Einstein flag manifolds \(K/T\) with an \({\mathbb S}^1\)-bundle such that the projection is biharmonic but not harmonic.
This article studies the biharmonicity of the projection \(\pi : (P,g) \to (M,h)\) where \((P,g)\) is an \({\mathbb S}^1\)-bundle over a compact Riemannian manifold \((M,h)\) and \(\pi\) is a Riemannian submersion.
First, it is proved that for a base manifold with non-positive Ricci curvature, when \(\pi\) is biharmonic, if the tension field of \(\pi\) is divergence-free then it must be parallel.
The cases of a weakly stable Einstein base manifold and of an irreducible compact Hermitian symmetric space are studied in more details and characterisations of the biharmonicity of the bundle projection are given.
This article closes with explicit examples of Kähler-Einstein flag manifolds \(K/T\) with an \({\mathbb S}^1\)-bundle such that the projection is biharmonic but not harmonic.
Reviewer: Eric Loubeau (Brest)
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