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integration of differential forms

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Integration theory

Differential geometry

differential geometry

synthetic differential geometry

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Idea

Here we discuss the integration of a differential form (possibly twisted in some way) on a topological manifold (possibly with additional structure) over an appropriately structured submanifold (or formal linear combination thereof).

Description

See differential form for basic definitions.

Integration of top-dimension pseudoforms (pseudoforms to measures)

Let XX be an nn-dimensional topological manifold, and let ω\omega be a continuous nn-pseudoform on XX. Suppose that XX is paracompact and Hausdorff, so that we may find a locally finite cover of XX with a subordinate partition of unity and a continuous coordinate chart on each patch. (When XX is differentiable, or even smooth, then these may also be chosen to be differentiable or smooth, which may be convenient but is not necessary.) Then ω\omega defines a measure on XX as follows:

  • On each coordinate patch UU, fix the orientation given by the coordinates to turn ω\omega into an untwisted nn-form ω^\hat{\omega}; then write ω^\hat{\omega} in coordinates as

    ω^=ω Udx 1dx n. \hat{\omega} = \omega_U \wedge \mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^n .

    In this situation, it is convenient also to write

    ω=ω Udx 1dx n; \omega = \omega_U \mathrm{d}x^1 \cdots \mathrm{d}x^n ;

    in other words, we interpret dx 1dx n\mathrm{d}x^1 \cdots \mathrm{d}x^n as the absolute value of dx 1dx n\mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^n.

  • The coordinates on UU define a diffeomorphism between UU and an open subset of R n\mathbf{R}^n that we'll also call UU; so use the latter formula to interpret

    (1) Uω= Uω U(x 1,,x n)dx 1dx n, \int_U \omega = \int_U \omega_U(x^1,\ldots,x^n)\, \mathrm{d}x^1 \cdots \mathrm{d}x^n ,

    where the right-hand side is now interpreted in the usual way as as integral with respect to Lebesgue measure.

  • Using the partition of unity, write

    ω= Uw Uω U, \omega = \sum_U w_U \omega_U ,

    where w Uw_U is a weight function defined on UU and ω U\omega_U is the restriction of ω\omega to UU. Then we have

    Xω= U Uw U(x 1,,x n)ω U(x 1,,x n)dx 1dx n, \int_X \omega = \sum_U \int_U w_U(x^1,\ldots,x^n) \omega_U(x^1,\ldots,x^n)\, \mathrm{d}x^1 \cdots \mathrm{d}x^n ,

    or more generally,

    Eω= U Uχ E(x 1,,x n)w U(x 1,,x n)ω U(x 1,,x n)dx 1dx n \int_E \omega = \sum_U \int_U \chi_E(x^1,\ldots,x^n) w_U(x^1,\ldots,x^n) \omega_U(x^1,\ldots,x^n)\, \mathrm{d}x^1 \cdots \mathrm{d}x^n

    for EE a measurable subset of XX and χ E\chi_E the characteristic function of EE.

A priori, this definition depends not only on the particular coordinate patches chosen but also on the partition of unity chosen to go with them. Furthermore, the defintion could be done just as easily (perhaps even more easily) for something other than an nn-pseudoform. But the (perhaps surprising) fact that justifies it all is this:

Theorem

When ω\omega is an nn-pseudoform, the definition of Eω\int_E \omega is independent of the coordinates and partition chosen. Furthermore, the map from nn-pseudoforms to measures is linear.

Note that, if ω\omega were an nn-form instead of a pseudoform, then the definition would depend on the orientation of the coordinates chosen. We could fix that by using the absolute value ω U{|\omega_U|} in place of ω U\omega_U in (1) and the following equations, but then the map from forms to measures would not be linear.

Measures to pseudoforms

It may also be enlightening to consider how to go back from a measure to an nn-pseudoform. If ω\omega is an absolutely continuous Radon measure on XX, then it defines an nn-pseudoform (which we may also call ω\omega) as follows:

  • Given a point aa, choose one of the two local orientations at aa.
  • Given nn linearly independent vectors (v 1,,v n)(v_1,\ldots,v_n) at aa, develop them into a coordinate system on a neighbourhood UU of aa.
  • For sufficiently large natural number kk, the coordinate cube C kC_k of points with coordinates in [0,1/k] n[0,1/k]^n exists (lies within UU).
  • Let LL be lim kk nω(C k)\lim_{k \to \infty} k^n \omega(C_k).
  • If the coordinate system on UU is positively oriented at aa, then let ω(v 1,,v n)\omega(v_1,\ldots,v_n) be LL; if the coordinate system on UU is negatively oriented at aa, then let ω(v 1,,v n)\omega(v_1,\ldots,v_n) be L-L.
  • Extend the definition to nn arbitrary vectors by continuity (which necessarily maps a linearly dependent tuple of vectors to zero).

Again, this definition is independent of the coordinate system chosen (as long as it extends the given vectors); or if that's not true, then we messed up and need to add further restrictions to the absolutely continuous Radon measure ω\omega. The definition is not independent of the orientation chosen, of course; thus we get a pseudoform rather than an untwisted form. You might try to ignore the orientation and take ω(v 1,,v n)\omega(v_1,\ldots,v_n) to be LL always, but that does not define an exterior form, as is most easily seen if two vectors are switched (which does not change LL). Instead, this would define an absolute differential form (which is equivalent to a pseudoform when, as here, the degree equals the dimension).

Integration of more general forms

One can integrate forms other than nn-pseudoforms, of course, but only over certain structures within the manifold XX. Specifically, if RR is a pp-dimensional submanifold of XX (that is a pp-dimensional manifold UU equipped with a map R:UXR\colon U \to X), then we would like to integrate pp-forms or pp-pseudoforms (defined on XX) over RR. Here is how we do this:

  • We may integrate a pp-form η\eta over RR if RR is oriented, that is if UU is oriented. We pull back η\eta from XX to UU, then use the orientation on UU to turn η\eta into a pp-pseudoform, which we can then integrate on the pp-dimensional manifold UU.

  • We may integrate a pp-pseudoform η\eta over RR if RR is pseudooriented, that is if it is equipped with a map that, for each point aa on UU, takes a local orientation of XX at R(a)R(a) to a local orientation of UU at aa, continuously in aa and taking opposite orientations to opposite orientations. Then locally, we turn η\eta into a pp-form on XX using a local orientation on XX, pull that back to UU, and use the corresponding local orientation on UU to turn that back into a pp-pseudoform, which we can then integrate on UU.

Thus, while integration of nn-pseudoforms is the most basic, integration of general pp-forms is actually a bit simpler than integration of general pp-pseudoforms. Integration of other twisted or vector-valued forms can also be done, again given appropriate structure on RR.

Note that, if XX is thought of a submanifold of itself, then it has a natural pseudoorientation that takes each local orientation to itself, and so we recover the original definition of integration of nn-pseudoforms on XX. Also, if XX is an oriented manifold, then it may be viewed as an oriented submanifold of itself, giving a definition of integration of nn-forms on XX.

To integrate on unoriented submanifolds of arbitrary dimension, use absolute differential forms.

Integration on more general domains

One often sees the definition of integration given for parametrised submanifolds, that is submanifolds where UU is an open subset of R p\mathbf{R}^p. This amounts to a combination of the concepts above, with the two uses of UU (as a coordinate patch in XX or as the source of a submanifold of XX) identified. The theorem that the integral of an nn-pseudoform on XX is independent of the coordinates chosen now becomes a theorem that the integral of a parametrised submanifold is independent of the parametrisation (up to some details about orientation), which in the end returns the result that one can integrate forms over arbitrary submanifolds (given an orientation or pseudoorientation as above).

We can also integrate on a formal linear combination of submanifolds, which is handy since these are the chains of de Rham homology?. This is straightforward:

iα iU iω iα i U iω. \int_{\sum_i \alpha_i U_i} \omega \coloneqq \sum_i \alpha_i \int_{U_i} \omega .

Properties

References

An exposition with an eye towards applications in physics is section 3 of

Revised on July 11, 2013 23:28:20 by Urs Schreiber (82.113.106.241)