Calibrated geometries in quaternionic Grassmannians (Q917064)

The method of calibrated geometries is applied to the invariant differential forms on the quaternionic Grassmannians \(G_ p({\mathbb{H}}^{p+q})\) of p-dimensional right \({\mathbb{H}}\)-subspaces of the \(p+q\)-dimensional right quaternionic vector space \({\mathbb{H}}^{p+1}\) to show that the sub-Grassmannians \(G_ p(E)\) of p-dimensional right \({\mathbb{H}}\)-subspaces of a \(p+r\)-dimensional right \({\mathbb{H}}\)-subspace E of \({\mathbb{H}}^{p+q}\) are volume minimizing in their real homology classes. Moreover, it is shown that any volume minimizing submanifold in such homology classes are congruent to a sub-Grassmannian.