Calibrated geometry
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mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
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 Edmond Bonan (born 27 January 1937 in Haifa, Mandatory Palestine) is a French mathematician, known particularly for his work on special holonomy. Biography After completing his undergraduate studies ...
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 Edmond Bonan (born 27 January 1937 in Haifa, Mandatory Palestine) is a French mathematician, known particularly for his work on special holonomy. Biography After completing his undergraduate studies ...
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 Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor *Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chromo ...
. 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 In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
, suitably normalized powers of the
Kähler form Kähler may refer to: ;People * Alexander Kähler (born 1960), German television journalist * Birgit Kähler (born 1970), German high jumper *Erich Kähler (1906–2000), German mathematician *Heinz Kähler (1905–1974), German art historian and a ...
are calibrations, and the calibrated submanifolds are the complex submanifolds. This follows from the Wirtinger inequality. * On a
Calabi–Yau manifold In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstri ...
, 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

*. *. * . * . * . * . * . * . * . * . *. * . * . * . * . * . * . * . * . * . * {{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