Uniqueness of volume-minimizing submanifolds calibrated by the first Pontryagin form

One way to understand the geometry of the real Grassmann manifold parameterizing oriented -dimensional subspaces of is to understand the volume-minimizing subvarieties in each homology class. Some of these subvarieties can be determined by using a calibration. In previous work, one of the authors calculated the set of -planes calibrated by the first Pontryagin form on for all , and identified a family of mutually congruent round -spheres which are consequently homologically volume-minimizing. In the present work, we associate to the family of calibrated planes a Pfaffian system on the symmetry group , an analysis of which yields a uniqueness result; namely, that any connected submanifold of calibrated by is contained in one of these -spheres. A similar result holds for -calibrated submanifolds of the quotient Grassmannian of non-oriented -planes.