Almost quaternionic structures on quaternionic kaehler manifolds

In this work, we consider a Riemannian manifold M with an almost quaternionic structure V defined by a three-dimensional subbundle of (1, 1) tensors F, G, and H such that {F, G, H} is chosen to be a local basis for V. For such a manifold there exits a subbundle ℋ(M) of the bundle of orthonormal frames ����(M). If M admits a torsion-free connection reducible to a connection in ℋ(M), then we give a condition such that the torsion tensor of the bundle vanishes. We also prove that if M admits a torsion-free connection reducible to a connection in ℋ(M), then the tensors F², G², and H² are torsion-free, that is, they are integrable. Here F, G, H are the extended tensors of F, G, and H defined on M. Finally, we show that if the torsions of F², G² and H² vanish, then M admits a connection with torsion which is reducible to ℋ(M), and this means that F², G², and H² are integrable.