Biharmonic maps on principal $$G$$-bundles over complete Riemannian manifolds of nonpositive Ricci curvature.(English)Zbl 1472.58013

This article considers principal $$G$$-bundles, equipped with a Sasaki-type metric, over a Riemannian manifold and the canonical projection $$\pi$$, which is then a Riemannian submersion.
The problem investigated is to find conditions such that $$\pi$$ biharmonic implies $$\pi$$ harmonic.
The first theorem proved is that if the principal $$G$$-bundle is compact and has non-positive Ricci curvature then $$\pi$$ biharmonic implies $$\pi$$ harmonic. The reader will notice in the proof that the one-form $$\alpha$$ defined by Equation (3.7) is not quite well-defined on the base manifold unless the tension field of $$\pi$$ is actually basic. This should then be compared with [C. Oniciuc, An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 48, No. 2, 237–248 (2002; Zbl 1061.58015)].
The second set of conditions is to assume the principal $$G$$-bundle is non-compact complete with non-positive Ricci curvature and the energy and bienergy of $$\pi$$ are finite. Then $$\pi$$ biharmonic implies $$\pi$$ harmonic. The proof of this second theorem is really only a rehash of N. Nakauchi et al. [Geom. Dedicata 169, 263–272 (2014; Zbl 1316.58012)].

MSC:

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

Citations:

Zbl 1061.58015; Zbl 1316.58012
Full Text:

References:

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