Calibrated geometry

In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold ("M","g") of dimension "n" equipped with a differential "p"-form "φ" (for some 0 ≤ "p" ≤ "n") which is a calibration, meaning that:
Set "G""x"("φ") = { "ξ" as above : "φ"|"ξ" = vol"ξ" }. (In order for the theory to be nontrivial, we need "G""x"("φ") to be nonempty.) Let "G"("φ") be the union of "G""x"("φ") for "x" in "M".
The theory of calibrations is due to R. Harvey and B. Lawson and others. Much earlier (in 1966) Edmond Bonan introduced G2-manifold and Spin(7)-manifold, constructed all the parallel forms and showed that those manifolds were Ricci-flat. Quaternion-Kähler manifold were simultaneously studied in 1967 by Edmond Bonan and Vivian Yoh Kraines and they constructed the parallel 4-form.
Calibrated submanifolds.
A "p"-dimensional submanifold "Σ" of "M" is said to be a calibrated submanifold with respect to "φ" (or simply "φ"-calibrated) if T"Σ" lies in "G"("φ").
A famous one line argument shows that calibrated "p"-submanifolds minimize volume within their homology class. Indeed, suppose that "Σ" is calibrated, and "Σ" ′ is a "p" submanifold in the same homology class. Then
where the first equality holds because "Σ" is calibrated, the second equality is Stokes' theorem (as "φ" is closed), and the inequality holds because "φ" is a calibration.