Abstract A compatible almost complex structure on a Riemannian manifold can be considered as a smooth map J from the manifold into its twistor space endowed with a natural Riemannian metric induced by the metric of the base manifold or as a self-map J of the unit tangent bundle endowed with the Sasaki metric. The conditions under which the map J is harmonic have been found in Davidov et al. (2018) in the four-dimensional case. In the present paper, we discuss the problem when the map J is harmonic and show that if J is harmonic, so is J , but not vice versa. Also, we obtain geometric conditions under which J is a harmonic map in the case when the Riemannian manifold is of dimension four.

中文翻译：

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定义单位切丛调和映射的几乎 Hermitian 结构

摘要 黎曼流形上的相容近复结构可以被认为是从流形到其扭曲空间的光滑映射 J，该映射具有由基流形的度量引起的自然黎曼度量或单位切线的自映射 J。赋予 Sasaki 度量的束。Davidov 等人已经发现了映射 J 调和的条件。（2018）在四维情况下。在本文中，我们讨论了映射 J 调和时的问题，并表明如果 J 是调和的，则 J 也是，反之则不然。此外，我们还获得了在黎曼流形为四维的情况下 J 是调和映射的几何条件。