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 &phi; (for some 0 ≤ p ≤ n) which is a calibration, meaning that: Set Gx(&phi;) = { &xi; as above : &phi;|&xi; = vol&xi; }. (In order for the theory to be nontrivial, we need Gx(&phi;) to be nonempty.) Let G(&phi;) be the union of Gx(&phi;) for x in M.
 * &phi; is closed: d&phi; = 0, where d is the exterior derivative
 * for any x ∈ M and any oriented p-dimensional subspace &xi; of TxM, &phi;|&xi; = &lambda; vol&xi; with &lambda; ≤ 1. Here vol&xi; is the volume form of &xi; with respect to g.

The theory of calibrations is due to R. Harvey and B. Lawson and others. Much earlier (in 1966) Edmond Bonan introduced G2-manifolds and Spin(7)-manifolds, constructed all the parallel forms and showed that those manifolds were Ricci-flat. Quaternion-Kähler manifolds 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 &Sigma; of M is said to be a calibrated submanifold with respect to &phi; (or simply &phi;-calibrated) if T&Sigma; lies in G(&phi;).

A famous one line argument shows that calibrated p-submanifolds minimize volume within their homology class. Indeed, suppose that &Sigma; is calibrated, and &Sigma;&thinsp;&prime; is a p submanifold in the same homology class. Then


 * $$\int_\Sigma \mathrm{vol}_\Sigma = \int_\Sigma \varphi = \int_{\Sigma'} \varphi \leq \int_{\Sigma'} \mathrm{vol}_{\Sigma'}$$

where the first equality holds because &Sigma; is calibrated, the second equality is Stokes' theorem (as &phi; is closed), and the inequality holds because &phi; is a calibration.

Examples

 * On a Kähler manifold, suitably normalized powers of the Kähler form are calibrations, and the calibrated submanifolds are the complex submanifolds. This follows from the Wirtinger inequality.
 * On a Calabi–Yau manifold, the real part of a holomorphic volume form (suitably normalized) is a calibration, and the calibrated submanifolds are special Lagrangian submanifolds.
 * On a G2-manifold, both the 3-form and the Hodge dual 4-form define calibrations. The corresponding calibrated submanifolds are called associative and coassociative submanifolds.
 * On a Spin(7)-manifold, the defining 4-form, known as the Cayley form, is a calibration. The corresponding calibrated submanifolds are called Cayley submanifolds.