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 Gx(φ) = { ξ as above : φ|ξ = volξ }.
(In order for the theory to be nontrivial, we need Gx(φ) to be nonempty.)
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.
A famous one line argument shows that calibrated p-submanifolds minimize volume within their homology class.