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I think the answer is no, for ##i: S \rightarrow \mathbb R^n ## the inclusion/restriction, and ##w## any form, we have that ##i_{*}dw=d(i_{*}w )## , but the homology of ##\mathbb R^n## is trivial(i.e., every closed form is exact), so that we can write ##w= d(\alpha)##, for some form ##\alpha##. Substituting back in above: ##i_{*}(d(d\alpha))=0 ##, since ##d^2=0##, so that any EDIT closed form in ##\mathbb R^n## restricts to a trivial form.

So as a counterexample, take any nonzero form on a submanifold. Is this correct ?

A problem seems to be that ##w## may not be closed. So this only shows closed forms restrict to closed forms.

(Any examples of non-closed forms in ##\mathbb R^n##?)

Just curious as to how to answer this. Maybe using the inclusion/restriction map? What if we showed the restriction/inclusion map is/ (is not) onto?

We have: ##i^{*}: T_p \mathbb R^n → T_p^* S## as a map between the respective cotangent bundles, given by:

##i_{*}w(X_p)= w(di)(X_p)##

Where di is the pushforward/tangent map of the inclusion. How do I take it from here?

(Let me use R^n instead of ##\mathbb R^n## to simplify )

I think we need to use properties of the inclusion map of a submanifold S into the ambient manifold re the tangent space. Doesn't then T_p S inject into T_p R^n (meaning T_pS is a subspace of T_p R^n as (del/delX_1,...,del/delX_n)→ (del/delX_1,....,del/delX_n,0,0,...0))?

I think I am missing some basic linear algebra here, but not sure on what.

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# Is Every Diff. Form on a Submanifold the Restriction of a Form in R^n?

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