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:

• φ is closed: dφ = 0, where d is the exterior derivative
• for any x ∈ M and any oriented p-dimensional subspace ξ of T_xM, φ|_ξ = λ vol_ξ with λ ≤ 1. Here vol_ξ is the volume form of ξ with respect to g.

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-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.