(this question did not get any answers on math.SE, so I am reposting it here)

Let $M$ be an $n$-dimensional manifold. Then the space of currents $\mathcal D^k(M)$ of degree $k$ on $M$ is the space of continuous linear functionals on the space of test $n-k$-forms. Two typical examples of currents are $$ \mathcal{F}(\omega) = \int_\Gamma \omega $$ where $\Gamma$ is a $n-k$-chain, and $$ \mathcal{G}(\omega) = \int_M \eta \wedge \omega $$ where $\eta$ is a $k$-form.

One can extend the action of Lie derivative from forms to currenst simply by $$ L_X \mathcal{H}(\omega) = \mathcal{H}(L_X \omega) $$ So the algebra of differential operators on $M$ acts on the space of $k$-currents.

There are also currents that are obtained by contraction of a $k$-vector with the test form, but I am not sure whether they cannot be obtained by applying differential operators to currents of the two kinds mentioned above.

Is it true that the examples of currents above generate the space of currents as a module over the ring of differential operators, i.e. any current can be obtained from a current like $\mathcal{F}$ or $\mathcal{G}$ by successive applications of Lie derivatives? If no, can the statement be repaired by adding more closure operations/generators? Or is it a completely wrong attitude, and a general current is a significantly more "singular" object?

  • $\begingroup$ Could you explain what you mean by "currents obtained by contraction of a $k$-vector with the test form"? If $\omega$ is a $k$-form and $\zeta$ a $k$-vector, $\langle \omega, \zeta \rangle$ is a smooth function. Maybe you want to fix a volume form $v$ and take the integral $\int_M \langle \omega, \zeta \rangle v$ ? $\endgroup$ – Qfwfq Oct 1 '15 at 15:30
  • $\begingroup$ Let $\zeta_0$ be a $k$-vector at $x\in M$. Can the current $\delta_{p,\zeta}: \omega \mapsto \langle \omega |_p \, ,\,\zeta_0 \rangle$ be realized as you claim? Or is it a weak limit of the currents in your question? $\endgroup$ – Qfwfq Oct 1 '15 at 15:36
  • $\begingroup$ @Qfwfq: what I mean is the current $T$ such that $T \omega$ is the value of $\omega$ on fixed $k$-vector $\zeta \in \Lambda^k_x M$ at a point $x$. Yes, $\delta_{p,\zeta}$ is an example of what I mean. I don't know if it can be realized as an application of differential operators to currents like $\mathcal F$ and $\mathcal G$ $\endgroup$ – Ivan Bodhidharma Oct 1 '15 at 15:50
  • $\begingroup$ [I made typo: of course I meant $p\in M$, not $x$] $\endgroup$ – Qfwfq Oct 1 '15 at 16:03

It may be helpful to look at a simple example. $\newcommand{\bR}{\mathbb{R}}$ Assume for simplicity that $M=\bR^n$. Fix a Radon measure $\mu$ on $\bR^n$. It defines a $0$-dimensional current on $\bR^n$. (I define the dimension of a current to be $n-\deg$.) If $\mu$ is not absolutely continuous with respect to the Lebesgue measure, then it cannot have the form $\mathcal{G}$. Also, if the support of $\mu$ is very complicated, this $0$-current is not given by the integration along a $0$-chain.

Also, it is not clear to me why there would exist a locally integrable function $f$ on $\bR^n$ and a differential operator $L$ such that $L f=\mu$ in the sense of distributions.

However, the flat currents seem to fit your description. (See these notes and the references therein for details.)


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