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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

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

Citations:

Zbl 0553.00001