## Abstract

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 F^{2}, G^{2}, and H^{2} 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 F^{2}, G^{2} and H^{2} vanish, then M admits a connection with torsion which is reducible to ℋ(M), and this means that F^{2}, G^{2}, and H^{2} are integrable.

Original language | English |
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Pages (from-to) | 1-13 |

Number of pages | 13 |

Journal | Bulletin of the Malaysian Mathematical Sciences Society |

Volume | 38 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 2015 |

### Bibliographical note

Publisher Copyright:© Malaysian Mathematical Sciences Society and Universiti Sains Malaysia 2014.

## Keywords

- Almost complex structure
- Almost quaternionic structure
- Subbundle
- Torsion tensor