A System of Pde for Calibrated Geometries

Given a real vector space V equipped with an Euclidean metric, (after rescaling) any p-form φ ∈ V p V ∗ defines a calibration on V . This note identifies an exterior differential system whose integral submanifolds are precisely the critical submanifolds of the calibration. In particular, calibrated submanifolds are necessarily integral submanifolds of the system. The result is extended to calibrations on G-manifolds, G a Lie subgroup of the special orthogonal group.