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 in the sense that
- φ is closed: dφ = 0, where d is the exterior derivative
- for any x ∈ M and any oriented p-dimensional subspace ξ of TxM, φ|ξ = λ volξ with λ ≤ 1. Here volξ is the volume form of ξ with respect to g.
Set Gx(φ) = { ξ as above : φ|ξ = volξ }. (In order for the theory to be nontrivial, we need Gx(φ) to be nonempty.) Let G(φ) be the union of Gx(φ) for x in M.
The theory of calibrations is due to R. Harvey and B. Lawson and others (see The History of Calibrations).
Read more about Calibrated Geometry: Calibrated Submanifolds, Examples, History of Calibrations (Geometric Measure Theory, Chapter 6, Section 6.5, Frank Morgan)
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