The Stable 4-dimensional Geometry of the Real Grassmann Manifolds

Authors

Weiqing Gu, Harvey Mudd CollegeFollow

Document Type

Article

Department

Mathematics (HMC)

Publication Date

1998

Abstract

In their fundamental paper Calibrated geometries in 1982 [HL], R. Harvey and B. Lawson, discussing future global applications, posed the problem of determining the subvarieties of the real Grassman manifolds GkRⁿ of oriented k-planes through the origin in Rⁿ which can be shown to be volume-minimizing in the homology classes by using as calibrations the invariant forms representing the universal Pontryagin classes.

We carry out this method here for the first Pontryagin form, using it at a calibration to determine volume-minimizing 4-dimensional cycles in all real Grassman manifolds. We also learn that their 4-dimensioncal geometry stabilizes immediately after the Grassmannian G₄R⁸.

Rights Information

© 1998 Duke University Press

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.

DOI

10.1215/S0012-7094-98-09306-1

Recommended Citation

Gu, Weiqing. "The stable 4-dimensional geometry of the real Grassmann manifolds." Duke Math. J. 93 (1998), no. 1, 155–178.