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One way to understand the geometry of the real Grassmann mani- foldGk(Rk+n) parameterizing orientedk-dimensional subspaces of Rk+n 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 4-planes calibrated by the rst Pontryagin form p1 on Gk(Rk+n )f or allk;n 4, and identied a family of mutually congruent round 4-spheres which are consequently homologically volume-minimizing. In the present work, we associate to the family of cal- ibrated planes a Pfaan system on the symmetry group SO(k + n; R), an analysis of which yields a uniqueness result; namely, that any connected sub- manifold of Gk(Rk+n) calibrated by p1 is contained in one of these 4-spheres. A similar result holds for p1-calibrated submanifolds of the quotient Grass- mannian G\k(R k+n )o f non-orientedk-planes.

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