# A dual theory to the theory of currents?

The k-currents are defined as dual space to the spaces of all smooth k-forms. (These monsters are used to work with the minimal k-surfaces.)

Assume I want to look at the generalized k-forms; they can be defined as certain functionals on the space of all smooth k-surfaces with boundary.

Did anybody consider such "generalized k-forms"?

In fact I am looking for such generalized 2-forms on $\mathbb R^n$ with values $so(n)$, this should be some kind of generalized curvature. (Instead of integral I should consider parallel translation around the boundary. If $n=2$, it gives me a sign-measure.) But I am also interested in the linear case.

• I'm not sure I understand what you're trying to do, but according to the motto "currents are to forms what distributions are to functions", currents already are a generalisation of smooth forms. On a closed n-manifold, a k-form $\alpha$ defines a $(n-k)$-current by integration : $D_\alpha : \beta \mapsto \int_M \alpha \wedge \beta$. IIRC, these currents are called "diffuse" and can be used to approximate every current. – Maxime Bourrigan Jul 18 '11 at 14:36
• To be pedantic: Maxime's comment only applies for oriented monifolds. Otherwise, it's a section of $\Lambda^{k}\otimes$(orientation line bundle) that defines a (n−k)-current by integration. – André Henriques Jul 18 '11 at 16:01
• Jenny Harrison has defined some complexes that are dual to the complexes of smooth differential forms, in a certain topology. I think that the tricky bit is exactly which topology to use. – Ben McKay Jul 18 '11 at 16:11
• @Maxime, I might be wrong, but it seems that dual space to the all smooth (n-k)-forms and to smooth k-submanifolds with boundary are different. – ε-δ Jul 19 '11 at 14:31
• Also: There exists a theory of generalized sections of every vector bundle (on any smooth manifold). – Johannes Hahn May 10 '14 at 19:38

## 2 Answers

There exists a notion of generalized form: cochains. These are linear functionals on a certain class of currents; the problem is that you would like to put on your class of currents a meaningfull topology (maybe induced by a norm) and you would also like this class to be stable under boundary.

Smooth $k-$surfaces with boundary have the second property, but not the first one. In order to put some topological structure on them, you have to enlarge the space and consider integer rectifiable currents, where you have a norm-induced topology, compactness theorems and stability under the boundary operator.

The cochains have been defined on flat currents (which contain the previous ones); you could look up for flat cochains into Federer's Geometric Measure Theory (4.1.19). They ultimately are functionals $\ell$ associated to a couple of measurable forms $(\alpha,\beta)$ of degree $k$ and $k-1$ so that for every flat current $T$ you have
$$\ell(T)=T(\alpha)+ \partial T(\beta)=T(\alpha)+T(d\beta)$$
They were also studied by Whitney, I believe you can find something in his Geometric Integration Theory (Chapter 9 and following).

I don't know if this was what you were looking for, but that's what I know about dualizing currents.

There is another notion of generalized form called charges. You can find its definition here : http://www.math.jussieu.fr/~depauw/preprints/dep-moo-pfe.pdf (they are functionals on the space normal currents with a distribution-like topology).

Cochains can be represented as $L^1$ forms with $L^1$ exterior derivative, whereas charges are represented as $\omega + d \eta$, where $\omega$ and $\eta$ are continuous forms. Maybe this kind of generalized form is best suited for your purposes.