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Let $M$ be a smooth $n$-dimensional manifold.

Please verify the following related conjectures about smooth extensions.

In the books From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave in particular here (see last sentence of this) and An Introduction to Manifolds by Loring W. Tu and here (about this), it seems we have the following

  1. Let $f: U \to \mathbb R$ be continuous and compactly supported with $U$ nonempty and open in $\mathbb R^n$. Then $\tilde{f} = f1_U+0 1_{U^c}$, extension by zero of the function $f$ is continuous.

    • I believe this is true because I believe this is proven as part of Tu Proposition 23.4 which goes: $\text{Assumptions} \implies \tilde{f} \ \text{continuous} \implies f \ \text{Riemann integrable}$.

  2. Merged with (5)

  3. Merged with (6)

  4. (Analogue of (1) for smooth): Let $f: U \to \mathbb R$ be smooth and compactly supported with $U$ nonempty and open in $\mathbb R^n$. Then $\tilde{f}$ is smooth.

    • I am not sure if this is true. Tu doesn't seem to give a sufficient condition to for smoothness of $\tilde{f}$ when we don't assume $f$ is compactly supported, either in Section 13 or the rest of the book, but I think this is what Madsen and Tornehave claim in Chapter 10.

  5. Let $\omega \in \Omega^k{U}$ with $U$ nonempty and open in $M$. Then $\tilde{\omega} = \omega1_U+0 1_{U^c}$, extension by zero of the form $\omega$ is not necessarily smooth.

    • $\Omega^k{U}$ is the $\mathbb R$-vector space and $C^{\infty}(U)=\Omega^0(U)$-module of smooth k-forms on $U$.

    • I believe this is true because I believe this is explained in Tu Section 13 for $0$-forms.

  6. Let $\omega \in \Omega^n{U}$ with $U$ nonempty and open in $M$. Then there exists $\rho$, a smooth bump function (defined below) at a point $q \in U$, such that $\hat{\omega} = \omega\rho1_U+0 1_{U^c}$ is smooth and agrees with $f$ on some neighborhood $W$ of $q$ in $U$ but not necessarily on the whole of $U$.

    • A bump function at a point $q \in U$ supported in $U$ is defined as a continuous nonnegative $\rho: M \to \mathbb R$ that is $1$ in some neighborhood $V$ of $q$ in $U$ with $\text{supp} \rho \subseteq U$ and exists by Tu Exercise 13.1, which I believe is applicable here.

    • I believe this is true because I believe this follows from Tu Proposition 18.8, an extension from the rule for $0$-forms, Tu Proposition 13.2, to $k$-forms, but is not quite equivalent. Tu Proposition 13.2 and Tu Proposition 18.8 are propositions about the existence of an $\hat{\omega}$ for a particular point in the domain of $f$. Here, I am making a conjecture without regard to a particular point: Since $U$ is nonempty, pick a point $q \in U$ and then apply Tu Proposition 18.8.

  7. (Analogue of (4) for top forms, i.e. n-forms): Let $\omega \in \Omega^n_c{U}$ with $U$ nonempty and open in $\mathbb R^n$. Then $\tilde{\omega} \in \Omega^n{U}$.

    • $\Omega^k_c{U}$ is the $\mathbb R$-vector space and $C^{\infty}(U)=\Omega^0(U)$-module of compactly supported smooth k-forms on $U$.

    • I believe this is true because I believe this I think this is what Madsen and Tornehave claim in Chapter 10.

  8. (1),(4) and (7) also hold if $U$ is instead empty, but the others above do not necessarily hold because the existence of a bump function assumes there is a point $q \in U$.

    • I believe this is true after some thought, but I have not found any references.
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    (4) is true because $f$ and $\overline f$ agree on $U$, and $f$ is assumed to be smooth. Outside $U$, $\overline f$ is clearly smooth since it is just constant. – lEm Jun 03 '19 at 09:21
  • @lEm Thanks! So it's the same argument as $f$ continuous and $\tilde f$ continuous in (1)? –  Jun 03 '19 at 23:36
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    Yes, as long as it is compactly supported. Actually the argument for differential forms is roughly the same, by working in a small open set. – lEm Jun 03 '19 at 23:48
  • So (1), (4) and (7) are pretty much the same for $U$ non empty? –  Jun 03 '19 at 23:49
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    Yes, indeed. They are fairly intuitive too. Afterall they are extension of smooth functions/forms, so you only need to check at the points on which discontinuity may happen, i.e. on the boundary $\partial U$, but they are constantly $0$ in a neighborhood of the boundary. – lEm Jun 04 '19 at 00:06
  • @lEm Thanks and what about (8) please? –  Jun 05 '19 at 05:55
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    It is not really that important to consider empty set. Because realistically we never need to extend a function/form defined on the empty set. But if you insist, note that there is a unique function on $\emptyset$, and it is vacuously continuous, smooth. Its extension is nothing but the zero function. – lEm Jun 05 '19 at 06:02
  • @lEm Okay so (1), (4) and (7) are vacuously true if $U$ becomes empty, but the others rely on the existence of bump functions which relies on non-empty $U$ and so the others are false for empty $U$? –  Jun 05 '19 at 06:04
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    No, you don't need bump function. Again, there is only one function and one k-form on empty set, they are vacuously smooth. The support is empty and so it is compact. In any case the extension $\tilde f=f1_U+01_{U^c}=0$, so it must be smooth. – lEm Jun 05 '19 at 06:07
  • @lEm I think I get it. The bump function arguments no longer hold, but you don't need bump functions as your comment demonstrates. Is this right? –  Jun 10 '19 at 05:54
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    Yes. ${}{} {} {} {} $ – lEm Jun 10 '19 at 05:57
  • @lEm Thanks! You can answer if you want. –  Jun 10 '19 at 05:59

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