BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, vol.38, no.1, pp.1-13, 2015 (SCI-Expanded)
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 O(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 (F-2) over tilde, (G(2)) over tilde, and (H-2) over tilde are torsion-free, that is, they are integrable. Here (F-2) over tilde, (G(2)) over tilde,(H-2) over tilde are the extended tensors of F, G, and H defined on M. Finally, we show that if the torsions of (F-2) over tilde, (G(2)) over tilde, and (H-2) over tilde vanish, then M admits a connection with torsion which is reducible to H(M), and this means that (F-2) over tilde, (G(2)) over tilde, and (H-2) over tilde are integrable.