Integration of differential forms

[Goldbeetle]

In my reference books differential forms are integrated by means of pullbacks. Actually, integrals of differential forms are DEFINED by means of pullbacks. In other words, the integration domain must have a parametrization. Since differential forms and their integrals are under regularity conditions independent of parametrization, as geometric objects, it should be possible to define integrals of differential forms independently of any parametrization and then, it should be proved that given a parametrization of the integration domain, one gets the same value for the integral. Does this make any sense? Is there any book that does develops the theory in this way?

 

[Landau]

Let M be an oriented m-manifold and $\alpha$ an m-form on M with compact support. Given an oriented atlas (U_i,k_i), you take a smooth partition of unity (f_i) subordinate to it, and define

$$\int_M\alpha:=\sum_i\int _{k_i(U_i)}(k_i^{-1})^*(f_i.\alpha).$$

It can be shown this is independent of the chosen chart and partition of unity.

 

[mathwonk]

A differential one form is a section of the cotangent bundle. OK, but it is integrated over a parametrized path. Suppose the path has no parametrization. Then even the orientation of the path is undefined, hence also the sign of the integral. If the path crosses itself, say an infinite number of times, then even the direction of the parametrization along the path is undetermined, hence the integral could have infinitely many different values.

So I think what you want (and what I also wanted at your age) is not possible. I was influenced by a philosophy that everything should be done in an "invariant" way, but this is somewhat nonsense in many concrete situations. Steep yourself in the definition and computation of these objects, and I think this desire will subside.

You might feel also that homology class is a nice invariant object, but how do you represent one? You have to choose a path, or a sequence of oriented segments. It helps to go through the details of a construction of homology to see how messy and explicit it is. We love to keep our hands clean and discuss math very abstractly, never calculating anything like an actual integral or an intersection number, but these things when they arise always require some explicit construction. The author often hides this from the reader or omits it entirely as an "exercise".E.g. you can represent anyone dimensional homology class on a torus, by a path that covers the entire torus. How do you decide which class it is without a parametrization? If you look at its trace, it looks like the whole torus, hence like a 2 dimensional class.

Try to search through explanations and calculations in math books for the nitty gritty part where they actually turn their backs and compute something explicit. People cover pages with abstract sheaf cohomology constructions and derived functors and injective resolutions ad infinitum, but then when they need to actually compute the cohomology of projective space, they usually write down some explicit cech cocycles on a concrete open cover.

I was brainwashed to think the concrete computations were the ugly part and the abstract stuff was the beautiful part, but those little secret computations with "t" and "1/t" or "t/(1+t^2)"in them, contain all the truth.

 

[Goldbeetle]

I agree with you. I'm no fanatic of "everything should be done invariant". It was just a question that dawn on me while studying.

 

[lavinia]

In my reference books differential forms are integrated by means of pullbacks. Actually, integrals of differential forms are DEFINED by means of pullbacks. In other words, the integration domain must have a parametrization. Since differential forms and their integrals are under regularity conditions independent of parametrization, as geometric objects, it should be possible to define integrals of differential forms independently of any parametrization and then, it should be proved that given a parametrization of the integration domain, one gets the same value for the integral. Does this make any sense? Is there any book that does develops the theory in this way?

Line integrals require oriented paths. Any path has two orientations and the line integral of a one form changes sign with change in orientation. Even the ordinary Riemann integral of a real function on a closed interval of numbers uses the orientation of the interval in the positive direction. If one reversed the orientation of the interval, the integral would change sign.

This same idea of orientation applies to integrals of higher dimensional differential forms.

So your question is how do you define orientated simplicial objects intrinsically in a space. I suspect that in a general topological space this is impossible. But in a smooth manifold one can choose a continuous section of the tangent bundle over a path.( Let's forget for the moment how you know that it is a path without using coordinates.)

But there is another problem. A Riemann integral seems to assume a measure on the parameter interval. This measure is just the parameter increment between two points. How does one choose a measure without a parameteriztion? My gut tells me that it could be done but I don't have any ideas.