A calibration form on a Riemannian manifold 
is a differential
p-form 
such that
1. 
is a closed form.
2. The comass of 
,
 |
(1)
|
defined using simple unit 
-vectors, equals 1.
A 
-dimensional submanifold
is calibrated when 
restricts to give the volume form.
It is not hard to see that a compact oriented calibrated submanifold 
without boundary minimizes its volume
among objects in its homology class. By Stokes'
theorem, if 
is a compact oriented submanifold without boundary representing the same homology class, then
 |
(2)
|
Since
 |
(3)
|
and
 |
(4)
|
it follows that the volume of 
is less than or equal to the volume of 
.
A simple example is 
on the plane, for which the lines 
are calibrated submanifolds.
In fact, in this example, the calibrated submanifolds
give a foliation. On a Kähler
manifold, the Kähler form 
is a calibration form. It is indecomposable
when the complex dimension is at least two,
but decomposable in complex
dimension one. For example, on
 |
(5)
|
the Kähler form is
 |
(6)
|
The Kähler form 
calibrates complex curves. More generally, 
calibrates the complex submanifolds
of complex dimension 
. Consequently, these complex submanifolds
are locally volume minimizing.
