One way to understand the geometry of the real Grassmann manifold Gk(Rk+n) parameterizing oriented k-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 first Pontryagin form p1 on Gk(Rk+n) for all k,n ≥4, and identified a family of mutually congruent round 4-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 SO(k+n, R), an analysis of which yields a uniqueness result; namely that any connected submanifold 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 Grassmannian Gk♮(Rk+n) of non-oriented k-planes.
© 2001 American Mathematical Society
Grossman, Daniel A.; Gu, Weiqing. "Uniqueness of volume-minimizing submanifolds calibrated by the first Pontryagin form." Trans. Amer. Math. Soc. 353 (2001), no. 11, 4319–4332.