After non-thorough literature search, it seems to me that there is no consensus on the usage of the terminology "(almost) hyper-Hermitian" vs "almost hyper-Kähler" vs "almost quaternion-Hermitian" etc.

Let $M$ be a manifold, consider the data of a Riemannian metric $g$ and three $g$-compatible almost complex structures $I$, $J$ and $K$ which satisfy the quaternionic relations $IJ = -JI = K$. When additionally $I$, $J$ and $K$ are parallel (hence integrable), everyone agrees to call this a "hyper-Kähler" structure.

When $I$, $J$ and $K$ are not assumed parallel or integrable, it seems to me that the most sensible terminology is to call this data an "almost hyper-Hermitian" structure, by analogy with the case of a single almost complex structure $I$. I'm a little surprised that some experts call this structure almost hyper-Kähler, e.g. Dominic Joyce in the book "compact manifolds with special holonomy" or Robert Bryant in the book "Geometry and Quantum Field Theory". By analogy with the case of a single almost complex structure $I$, I would only call it "almost hyper-Kähler" when the three associated Kähler forms are closed (though I am aware that this is actually sufficient to be hyper-Kähler, by a theorem of Hitchin). Another expert, Kaledin, just drops the "almost" and calls it "hyper-Hermitian" (see here) -weird.

Some seem to reserve "almost hyper-Hermitian" to the slightly weaker notion of a Riemannian manifold with a $Sp(n)$-structure, in other words the choice of $I$, $J$ and $K$ among the $S^2$-worth of almost complex structures that they generate is not fixed. I am not sure I understand the logic in this terminology. The way I see it is: just like $I$ is fixed in a complex vector space, $I$, $J$ and $K$ are fixed in an $\mathbb{H}$-module.

For the term "almost quaternion-Hermitian", which I think could sensibly be a mere alias for "almost hyper-Hermitian", authors appear to consistently reserve it for a Riemannian manifold with a $Sp(n)Sp(1)$-structure. Again I'm not sure I understand the logic in this terminology, but I suppose it is consistent with the common usage of "quaternion-Kähler" (which feels no more natural to me for the same reason, why isn't that just an alias for hyper-Kähler). But maybe I shouldn't have brought up (almost) quaternion Hermitian/Kähler structures, my initial interrogation was about (almost) hyper-Hermitian/Kähler structures.

Does anyone have insights on the question?


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