## Compatible complex structures on almost quaternionic manifolds.(English)Zbl 0933.53017

Let $$(M^{4n},Q)$$ be an almost quaternionic manifold. Assume that $$Q$$ is locally spanned by the almost complex structures $$J_1,J_1,J_3$$ such that $$J_1J_2=-J_2J_1=J_3$$. An almost complex structure $$I$$ on $$M^{4n}$$ is compatible with the almost quaternionic structure $$Q$$ if $$I=\sum^2_{\alpha=1} c_\alpha J_\alpha$$, locally, and $$\sum^3_{\alpha=1} c^2_\alpha=1$$.
The authors study the relationship between the 1-integrability of the almost quaternionic structure $$Q$$ and the existence of compatible complex structures. They obtain several very interesting results concerning the properties of the twistor space of an almost quaternionic manifold as well as some generalizations of the Penrose twistor constructions. In the case $$n\geq 2$$, there are two compatible complex structures $$I_1,I_2 \neq\pm I_1$$ on $$M^{4n}$$ if $$(M^{4n},Q)$$ is quaternionic. If $$n=1$$, a maximum principle for the angle function of two compatible almost complex structures follows. Then, an application to anti-self-dual manifolds is presented. There exists an almost complex structure on the twistor space $$Z$$ of an almost quaternionic manifold $$(M^{4n},Q)$$, and it is integrable if and only if $$Q$$ is quaternionic.
Reviewer: V.Oproiu (Iaşi)

### MSC:

 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C10 $$G$$-structures 53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
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### References:

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