Compatible complex structures on almost quaternionic manifolds

Abstract

On an almost quaternionic manifold (M4,, 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 Ii, I2 # ?I1 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 (Ii, I2) 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.