Almost Quaternionic Structures on Quaternionic Kaehler Manifolds

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 $\mathcal{{H}} (M)$ of the bundle of orthonormal frames $\mathcal{{O}}(M)$. If $M$ admits a torsion-free connection reducible to a connection in $\mathcal{{H}}(M)$, 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 $\mathcal{{H}}(M)$, then the tensors $\widetilde{F}^2$, $\widetilde{G}^2$, and $\widetilde{H}^2$ are torsion-free, that is, they are integrable. Here $\widetilde{F}$, $\widetilde{G}$, $\widetilde{H}$ are the extended tensors of $F$, $G$, and $H$ defined on $M$. Finally, we show that if the torsions of $\widetilde{F}^2$, $\widetilde{G}^2$ and $\widetilde{H}^2$ vanish, then $M$ admits a connection with torsion which is reducible to $\mathcal{{H}}(M)$, and this means that $\widetilde{F}^2$, $\widetilde{G}^2$, and $\widetilde{H}^2$ are integrable.