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# Compatible Complex Structures on Almost Quaternionic Manifolds

D. V. Alekseevsky, S. Marchiafava and M. Pontecorvo
Transactions of the American Mathematical Society
Vol. 351, No. 3 (Mar., 1999), pp. 997-1014
Stable URL: http://www.jstor.org/stable/117913
Page Count: 18
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## Abstract

On an almost quaternionic manifold (M4n,Q) we study the integrability of almost complex structures which are compatible with the almost quaternionic structure Q. If n≥ 2, we prove that the existence of two compatible complex structures I1,I2 ≠ ± I 1 forces (M4n,Q) to be quaternionic. If n = 1, that is (M4,Q)=(M4,[g],or) is an oriented conformal 4-manifold, we prove a maximum principle for the angle function $\langle I_{1},I_{2}\rangle$ of two compatible complex structures and deduce an application to anti-self-dual manifolds. By considering the special class of Oproiu connections we prove the existence of a well defined almost complex structure J on the twistor space Z of an almost quaternionic manifold (M4n,Q) and show that J is a complex structure if and only if Q is quaternionic. This is a natural generalization of the Penrose twistor constructions.

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