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Mathematics (HMC)

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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.


First published in Transactions of the American Mathematical Society 353 (2001), 4319-4332, published by the American Mathematical Society.

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© 2001 American Mathematical Society

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