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In Spivak's 'calculus on manifolds' Spivak defines a smooth differential $p$-form on a manifold $M \subset \mathbb{R}^n$ as an assignment $\omega$ of points $x \in M$ such that $\omega(x) \in \Lambda^p(M_x)$ where $M_x$ is the tangent space at $x$, and such that if $f$ is a coordinate chart containing $x$, $f^*\omega$ is smooth. So far so good, but what I don't understand is his next comment. Spivak claims every such $\omega$ can be written as $$\omega=\sum_{i_1<..<i_p} \omega_{i_1,..,i_p} dx_{i_1} \wedge .. \wedge dx_{i_p},$$ where the $\omega_{i_1,..,i_p}$ are functions defined only on $M$. My question is, he seems to be implying that for $y \in M$, $dx_1 \wedge ... \wedge dx_p (y) \in \Lambda^p(M_y)$, or at least that every element of $\Lambda^p(M_y)$ is a restriction of an element from $\Lambda^p(\mathbb{R^n})$. Could someone clarify what spivak could mean by this?

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  • $\begingroup$ If $M\subset \mathbb{R}^k$, then tangent spaces of $M$ are canonically identifiable to subspaces of $\mathbb{R}^n$. $\endgroup$ – Olivier Aug 23 '17 at 23:35
  • $\begingroup$ I understand that, but that still does not clarify everything to me $\endgroup$ – M. Van Aug 23 '17 at 23:42
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You ask a very good question. First of all, yes, $dx_1\wedge\dots\wedge dx_p$ by restriction gives an alternating $p$-tensor on the tangent space of $M$ at $y$, as you can feed in any $p$ vectors in $\Bbb R^n$ that you wish. More pedantically, if you consider the inclusion map $\iota\colon M\to\Bbb R^n$, you can take $\iota^*(dx_1\wedge\dots\wedge dx_p)$ to formalize the restriction. And, since (with increasing multi-indices) $dx_{i_1}\wedge\dots\wedge dx_{i_p}$ give a basis for $\Lambda^p(\Bbb R^n)$, they give, by restriction, a spanning set for $\Lambda^p(M_y)$.

But to give the global representation of any $p$-form as desired, I think, takes a bit more of an argument. By a partition of unity argument, it suffices to do this on an open set on which $M$ is a graph over a coordinate $k$-plane in $\Bbb R^n$. (That every point has such a neighborhood is an application of the inverse function theorem.) So, if on an open set $V_s\subset\Bbb R^n$, $M\cap V_s$ is represented as $(u,g(u))$, $u\in U_s$ in the $x_{j_1}\dots x_{j_k}$-plane, you can easily convince yourself that any $p$-form on $V_s$ is obtained by pulling back $p$-forms from this $k$-plane, and hence is a functional linear combination $\phi_s$ of $dx_{i_1}\wedge\dots\wedge dx_{i_p}$, where $i_\ell\in \{j_1,\dots,j_k\}$. This is, in particular, of our desired form (no pun intended). Now, just take a partition of unity $\rho_s$ subordinate to the collection $\{V_s\}$, and consider the form $\sum_s \rho_s\phi_s$, globally defined on $\Bbb R^n$.

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