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Let $M$ be a smooth oriented manifold of dimension $n$ and $T$ a current of dimension $k$ on $M$. Let $\phi:P\times M \to M$ be a proper smooth family of diffeomorphisms of $M$ (i.e. $P$ is a smooth manifold, $\phi$ is a smooth proper map, and each $\phi_p:=\phi(p, \cdot)$ is a diffeo of $M$) and $dp$ a smooth measure on $P$. Suppose that $\phi(\cdot,x)$ is a submersion for each $x \in M$.

I claim that the current $S:= \int_P \phi_{p*} T \, dp$ (i.e. $S(\alpha) = \int_P T(\phi_p^*\alpha) \, dp$ for $\alpha$ a $k$-form) is smooth, i.e. is given by integration against a smooth $(n-k)$-form. This should be a standard fact. I think I can give a proof but it is tedious. Does anyone know a reference?

I would be content with a reference covering the case where $T$ is given by integration over a smooth oriented submanifold, but surely this is an artificial restriction.

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  • $\begingroup$ If $P$ is a point and $\varphi=\mathrm{Id}_M$, then $S=T$ has no reason to be smooth, or am I missing something? $\endgroup$
    – Henri
    Apr 21, 2014 at 21:46
  • $\begingroup$ Henri--- your example does not satisfy the submersion condition. $\endgroup$
    – Joe Fu
    Apr 21, 2014 at 22:07
  • $\begingroup$ If $M = \mathbb R^n=P $, $P$ acts by translations, and the measure $dp$ is given by the smooth density $f$, then the operation reduces to convolution with $f$. Strictly speaking this example violates the properness condition, but if $f$ has compact support then it is OK (the properness condition can be relaxed to: $\phi$ is proper on $supp (dp) \times M$). $\endgroup$
    – Joe Fu
    Apr 21, 2014 at 22:12
  • $\begingroup$ indeed, I had missed the submersion assumption, my bad! $\endgroup$
    – Henri
    Apr 22, 2014 at 9:53
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    $\begingroup$ I think the result follows from the standard fact that smearing $T(x)$ with some $\rho(x,y)$, which is smooth in both arguments, gives something smooth. This can be made global using a partition of unity subordinate to a cover that is adapted to put the submersions $\phi(\cdot,x)$ in standard form. Since the basic fact is standard, it may not require a reference, but in any case you might try the standard books by Schwartz (distributions), de Rham (currents), Federer (geometric measure theory). $\endgroup$ Apr 22, 2014 at 12:27

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