- Calibrated geometry
In the mathematical field of
differential geometry , a calibrated geometry is aRiemannian 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: d"φ" = 0, where d is theexterior 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") = { "ξ" as above : "φ"|"ξ" = vol"ξ" }. (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 Harvey and Lawson.
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
:
where the first equality holds because "Σ" is calibrated, the second equality is
Stokes' theorem (as "φ" is closed), and the third equality holds because "φ" is a calibration.Examples
* On a
Kahler manifold , suitably normalized powers of theKahler form are calibrations, and the calibrated submanifolds are thecomplex submanifold s.
* On aCalabi-Yau manifold , the real part of a holomorphic volume form (suitably normalized) is a calibration, and the calibrated submanifolds arespecial Lagrangian submanifold s.
* 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 aSpin(7)-manifold , the defining 4-form is a calibration. The corresponding calibrated submanifolds are called Cayley submanifolds.References
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