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A calibrated manifold is a Riemannian manifold (M,g) of dimension n equipped with a differential p-form φ (for some 0 ≤ p ≤ n)

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 $T_x$M, φ|ξ = λ volξ with λ ≤ 1. Here volξ is the volume form of ξ with respect to g. Set $G_x$(φ) = { ξ as above : φ|ξ = volξ }. (In order for the theory to be nontrivial, we need $G_x$(φ) to be nonempty.) Let G(φ) be the union of $G_x$x(φ) for x in M.