Flat form

A measurable $ r $- dimensional differential form $ \omega $ on an open set $ R \subset E ^ {n} $ such that: 1) the co-mass (cf. Mass and co-mass) $ | \omega | _ {0} \leq N _ {1} $ for a given $ N _ {1} $; and 2) there exists an $ N _ {2} $ with

$$ \left |\; \int\limits _ {\partial \sigma ^ {r + 1 } } \omega \right | \leq N _ {2} | \sigma ^ {r+1} | $$

for any simplex $ \sigma ^ {r+ 1 } $ satisfying the following condition: There exists a measurable $ Q \subset R $, $ | R \setminus Q | _ {n} = 0 $, such that $ \omega $ is measurable on $ \sigma ^ {r+ 1 } $ and on any one of its boundaries $ \sigma ^ {r} $, making up $ \partial \sigma ^ {r+ 1 } $; moreover,

$$ | \sigma ^ {r + 1 } \setminus Q | _ {r + 1 } = 0,\ \ | \sigma ^ {r} \setminus Q | _ {r} = 0. $$

Here, $ | M | _ {s} $ denotes the $ s $- dimensional Lebesgue measure of the intersection of the set $ M $ with some $ s $- dimensional plane.

If $ X $ is an $ r $-dimensional flat cochain in $ R $, there exists a bounded $ r $-dimensional form $ \omega _ {X} $ in $ R $ which is measurable in any simplex $ \sigma ^ {r} $ with respect to the plane which contains $ \sigma ^ {r} $, and

$$ \tag{1 } X \sigma ^ {r} = \ \int\limits _ {\sigma ^ {r} } \omega _ {X} . $$

Also

$$ | \omega _ {X} | _ {0} = | X |,\ \ | \omega _ {dX} | _ {0} = | dX |, $$

where $ | X | $ is the co-mass of the cochain $ X $. Conversely, to any $ r $- dimensional flat form $ \omega $ in $ R $ there corresponds, according to formula (1), a unique $ r $- dimensional flat cochain $ X _ \omega $ for any simplex $ \sigma ^ {r} $ which satisfies the above condition; moreover,

$$ | X _ \omega | \leq N _ {1} ,\ \ | d X _ \omega | \leq N _ {2} . $$

The form $ \omega $ and the cochain $ X $ are called associated. Forms associated with the same cochain are equivalent, i.e. are equal almost-everywhere in $ R $, and comprise the flat representative.

There is a one-to-one correspondence between the $ n $-dimensional flat cochains $ X $ and the classes of equivalent bounded measurable functions $ \phi ( p) $, given by $ \omega _ {X} = \phi ( p) dp $, and

$$ \phi ( p) = \ \lim\limits _ {i \rightarrow \infty } \ \frac{X \sigma _ {i} }{| \sigma _ {i} | } , $$

where $ \sigma _ {1} , \sigma _ {2} ,\dots $ is a sequence of $ n $- dimensional simplices contracting towards the point $ p $ such that their diameters tend to zero, but such that

$$ \frac{| \sigma _ {i} | }{( \mathop{\rm diam} \sigma _ {i} ) ^ {n} } \geq \eta $$

for some value of $ \eta $, where $ | \sigma _ {i} | $ is the volume $ \sigma _ {i} $ for all $ i $.

Let $ \alpha ( p) $ be a measurable summable function in $ R $ whose values are $ r $- vectors; it is said to correspond to an $ r $- dimensional flat chain if

$$ \tag{2 } \int\limits _ { R } \omega _ {X} \cdot \alpha = X \cdot A $$

for all $ r $- dimensional flat cochains $ X $ ($ A $ is then called a Lebesgue chain). The mapping $ \alpha \rightarrow A $ is a linear one-to-one mapping of the set of equivalence classes of functions $ \alpha ( p) $ into the space of flat chains $ C _ {r} ^ \flat ( R) $; also, $ | A | = \int _ {R} | \alpha | _ {0} $, where $ | A | $ is the mass of the chain $ A $, (cf. Mass and co-mass) and $ | \alpha | _ {0} $ is the mass of the $ r $-vector $ \alpha ( p) $. In addition, the set of images of continuous functions $ \alpha $ is dense in $ C _ {r} ^ \flat ( R) $.

Formulas (1) and (2) generalize similar results for sharp forms and sharp cochains (cf. Sharp form); for instance, the differential of the flat form $ \omega _ {X} $, defined by the formula $ d \omega _ {X} = \omega _ {dX _ \omega } $, is also a flat form, and Stokes' theorem: $ \int _ {\partial \sigma } \omega = \int _ \sigma d \omega $ is valid for any simplex $ \sigma $; an $ r $- dimensional flat cochain is the weak limit of smooth cochains, i.e. cochains for which the associated forms $ \omega $ are smooth, etc.

References

[1]	H. Whitney, "Geometric integration theory" , Princeton Univ. Press (1957)