Calibrated geometry

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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 ≤ pn) which is a calibration, meaning that:

  • φ is closed: dφ = 0, where d is the exterior derivative
  • for any xM 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. Much earlier (in 1966) Edmond Bonan introduced G2-manifold and Spin(7)-manifold, constructed all the parallel forms and showed that those manifolds were Ricci-flat. Quaternion-Kähler manifold were simultaneously studied in 1967 by Edmond Bonan and Vivian Yoh Kraines and they constructed the parallel 4-form.

Calibrated submanifolds[edit]

A p-dimensional submanifold Σ of M is said to be a calibrated submanifold with respect to φ (or simply φ-calibrated) if TΣ lies in G(φ).

A famous one line argument shows that calibrated p-submanifolds minimize volume within their homology class. Indeed, suppose that Σ is calibrated, and Σ ′ is a p submanifold in the same homology class. Then

where the first equality holds because Σ is calibrated, the second equality is Stokes' theorem (as φ is closed), and the inequality holds because φ is a calibration.



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