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Fix an integer $k\ge 1$. Let $X$ be a smooth manifold. It is well-known, that a real valued function on $X$ is smooth if its restriction along all smooth maps $M\to X$ for manifolds of dimension $\le k$ is smooth. A similar assertion seems to be true for $p$-forms on $X$ as long as $p<k$. My question concerns the case $p=k$.

Is it true that a $k$-form on $X$ is smooth if its restrictions along all smooth maps from manifolds of dimension $\le k$ are smooth.

The real question behind this is to understand the sheaf $i_*i^{*} \Omega$ on the category of smooth manifolds, where $i$ is the inclusion of the full subcategory of manifolds of dimension $\le k$ and $\Omega$ denotes the de Rham complex.

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    $\begingroup$ By restriction of a form do you mean the induced section of the pullback bundle and not the pullback of the form? $\endgroup$ Sep 30, 2014 at 8:23
  • $\begingroup$ I am pretty sure that this is correct and the necessary work is done in baby Rudin. But if I remember correctly, you couldn't just quote Rudin to get the result, because his definition of forms is quite messy. $\endgroup$
    – Ben McKay
    Sep 30, 2014 at 10:14
  • $\begingroup$ So you claim that a function $f\colon \mathbb R^2\to\mathbb R$ is smooth if the composition $f\circ h$ is smooth for any smooth regular curve $h\colon \mathbb R\to \mathbb R^2$. I do not think it is true, there are counterexamples. $\endgroup$ Sep 30, 2014 at 16:12
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    $\begingroup$ @Anton It is true by Boman's theorem (J.Boman: Differentiability of a function and of its compositions with functions of one variable, 1967). You can also find the statement in "Kriegl, Michor: The convenient setting of global analysis, Chapter I, Section 3.4." Actually I believe that a (positive) answer to Ulrich's question is to be found there too. $\endgroup$ Sep 30, 2014 at 20:21
  • $\begingroup$ I really mean the restriction of forms, not the pull-back section of the pull-back bundle. Let me indicate where I see the problem. Consider a form on $\mathbb{R}^{2}$ of the form $f(x,y)dx$. We want to detect smoothness of $f$ using a curve $(x(t),y(t))$. We must require that $x^{\prime}\not=0$ in order to get a nontrivial restriction. So we can not use all curves to detect smoothness of $f$. $\endgroup$ Oct 1, 2014 at 10:34

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