Harvey, Reese Calibrated geometries. (English) Zbl 0565.53032 Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 797-808 (1984). [For the entire collection see Zbl 0553.00001.] A closed differential p-form on a Riemannian manifold whose comass is bounded from above by 1 is called a calibration. Calibrations determine a geometry of submanifolds, so that an oriented p-dimensional submanifold minimizes volume in its homology class if the calibration restricts to the volume form of the submanifold. This article surveys these various geometries, e.g. calibrated foliations, Kähler geometry, special Lagrangian geometry, exceptional geometries, double-point geometry, etc. in which calibrations are used. Some of these geometries have analogues to the Wirtinger inequality and Cauchy-Riemann equation. Reviewer: J.Hebda Cited in 8 Documents MSC: 53C40 Global submanifolds 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry 53C12 Foliations (differential geometric aspects) 32Q99 Complex manifolds 53C55 Global differential geometry of Hermitian and Kählerian manifolds Keywords:minimal submanifolds; calibration; Kähler geometry; Lagrangian geometry; Wirtinger inequality Citations:Zbl 0553.00001 PDFBibTeX XML