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Is every Riemannian submersion necessarily a Harmonic map? If not under what condition that is true?

The motivation: the linear part of a Riemannian submersion is the direct sum og an isometry and the zero operator

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I just realize that the answer is negative:

  • Radu Pantilie, Some remarks on harmonic Riemannian submersion, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, Nouvelle Série Tome 40 (88), No. 1 (1997) pp. 21–26 https://www.jstor.org/stable/43678588

We give two conditions each one necessary and sufficient for a Riemannian submersion to be harmonic. As an application, we show how it can be modified the metric on the total space such that the Riemannian submersion to become harmonic.

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  • $\begingroup$ @DavidRoberts I appreciate very much your kind help and edit $\endgroup$
    – Ali Taghavi
    Commented Sep 10 at 9:21

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