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, meaning that:
* ''φ'' is closed: d''φ'' = 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'' vol_''ξ'' with ''λ'' ≤ 1. Here vol_''ξ'' is the volume form of ''ξ'' with respect to ''g''.
Set ''G''_''x''(''φ'') = . (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''(''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-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.

Calibrated submanifolds

A ''p''-dimensional submanifold ''Σ'' of ''M'' is said to be a calibrated submanifold with respect to ''φ'' (or simply ''φ''-calibrated) if T''Σ'' 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

:$\int_\Sigma \mathrm_\Sigma = \int_\Sigma \varphi = \int_ \varphi \leq \int_ \mathrm_$

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.

Examples

* On a Kähler manifold, suitably normalized powers of the Kähler form are calibrations, and the calibrated submanifolds are the complex submanifolds. This follows from the Wirtinger inequality.
* On a Calabi–Yau manifold, the real part of a holomorphic volume form (suitably normalized) is a calibration, and the calibrated submanifolds are special Lagrangian submanifolds.
* On a G₂-manifold, both the 3-form and the Hodge dual 4-form define calibrations. The corresponding calibrated submanifolds are called associative and coassociative submanifolds.
* On a Spin(7)-manifold, the defining 4-form, known as the Cayley form, is a calibration. The corresponding calibrated submanifolds are called Cayley submanifolds.

References

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* {{citation , first = W. , last = Wirtinger , title = Eine Determinantenidentität und ihre Anwendung auf analytische Gebilde und Hermitesche Massbestimmung, journal = Monatshefte für Mathematik und Physik , volume = 44 , year = 1936 , pages = 343–365 (§6.5) , doi = 10.1007/BF01699328, s2cid = 121050865 .

Differential geometry
Riemannian geometry
Structures on manifolds