In the
mathematical field of
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, a calibrated manifold is a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
(''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
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
* 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)-manifold
In mathematics, a Spin(7)-manifold is an eight-dimensional Riemannian manifold whose holonomy group is contained in Spin(7). Spin(7)-manifolds are Ricci-flat and admit a parallel spinor. They also admit a parallel 4-form, known as the Cayley form, ...
s, constructed all the parallel forms and showed that those manifolds were Ricci-flat.
Quaternion-Kähler manifold In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is a subgroup of Sp(''n'')·Sp(1) for some n\geq 2. Here Sp(''n'') is the sub-group of SO(4n) con ...
s 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
:
where the first equality holds because ''Σ'' is calibrated, the second equality is
Stokes' theorem
Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
(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 submanifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a com ...
s. 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 submanifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...
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 a
Spin(7)-manifold
In mathematics, a Spin(7)-manifold is an eight-dimensional Riemannian manifold whose holonomy group is contained in Spin(7). Spin(7)-manifolds are Ricci-flat and admit a parallel spinor. They also admit a parallel 4-form, known as the Cayley form, ...
, 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