Quaternionic manifolds, also called almost hypercomplex, are defined by the existence of an ($\mathbb{R}$-linear) action of quaternions on each tangent space such that $I,J,K\in\mathbb{H}$ are global sections of $\operatorname{End} TX$.
Several classes of manifolds can be defined by the reduction of their structure group from $GL(n,\mathbb{R})$ to a subgroup:
- $O(n)$: Riemannian manifold ($\exists g$ symmetric positive);
- $Sp(2(n/2),\mathbb{R})$: almost symplectic manifold ($\exists\omega$ nondegenerate $2$-form);
- $GL(n/2, \mathbb{C})$: almost complex manifold ($\exists J\in\operatorname{End}(TX)$ with $J^2=-1$)
- $U(n/2)$: almost Hermitian manifold ($\exists\omega,J$ as above)
- Can we put a group here?: almost hypercomplex manifold ($\exists I, J, K\in\operatorname{End}(TX)$ with $I^2=J^2=K^2=IJK=-1$)
For completeness, let me recall the following. A reduction to $G\subset GL(n,\mathbb{R})$ is a section of the quotient by $G$ of the tangent frame bundle, namely a (continuous) choice at each point of a "copy of $G$" ($G$-torsor) among bases of the tangent space.
For example a reduction to $O(n)$ is a (continuous) choice of which bases of tangent space are orthonormal. Similarly a reduction to $GL(n/2, \mathbb{C})$ is a choice of which bases are compatible with the almost complex structure $J$.