**EDIT:** Let $M$ and $N$ are two smooth manifold and suppose $N$ is compact but $M$ is not necessarily compact. For my purpose, I just need to consider the case $M=\mathbb R \times S^1$ or $\mathbb R \times [0,1]$.

Thanks to David's comments that help me a lot, the problem I really concern is just the following.

**Question:**

- Now that we know if $M$ is non-compact then $C^\infty(M,N)$ is modeled on spaces $C^\infty_c(M,N)$. Since we know $W^{k,p}_0(\Omega)$ is the closure of $C^\infty_c(\Omega) $ in $W^{k,p}(\Omega)$ for $\Omega\subset \mathbb R$, in analogy, can we say $W^{k,p}(M,N)$ is modeled on spaces $W^{k,p}_0(M,f^*TN)$? Any reference?
- Can we consider $W^{k,p}_0(M,f^*TN)$ as a completion (in some sense?) of $C^\infty_c(M,f^*TN)$, the spaces of smooth sections with compact support of pullback bundles along $f:M\to N$?

Previous post:

**(1)** Can we define $W^{1,p}_0(M,N)$
in a similar manner that $W^{1,p}_0(\mathbb R)$ is defined by the completions of $C^\infty_0$ or $C^\infty_c$?

For example, Floer in his paper (see Definition 2.1) actually discusses (roughly) $\mathcal W:=W_{loc}^{k,p}(\mathbb R \times [0,1], N)$ with a topology given by open sets as follows: $ \mathcal O_{u,\rho,\epsilon} =\{ v\in \mathcal W \mid v= \exp_u \xi ~\text{on}~ [-\rho,\rho]\times[0,1], ~\text{and}~ ||\xi||_{W^{k,p}} <\epsilon \} $ On the other hand, Audin and Damian in their book (see Definition 8.2.2) consider Banach manifolds $\widetilde {\mathcal W} =W^{k,p}(\mathbb R \times [0,1], N)$ (actually $[0,1]$ should be replaced by $S^1$) in a quite different manner. Here open sets are the space of maps of the form $v=\exp_u \xi$ where $\xi \in W^{k,p}(\mathbb R \times [0,1], N)\equiv W_0^{k,p}(\mathbb R \times [0,1], N)$ and where $u$ is smooth and converges in some decay at the infinity.

Heuristically, $\mathcal W$ is like the completion of $C_c^\infty$ while $\widetilde {\mathcal W}$ is like that of $C_0^\infty$.

**(2)** Is $\mathcal W$ the same as $\widetilde{\mathcal W}$? Which one could be a better candidate for the definition of $W^{k,p}_0(M,N)$? Are they both Banach manifolds as in the case $M$ is compact?

- Recently I notice that when considering (infinite-dimensional) manifolds of mapping, say $C^\infty(M,N)$, we usually require $M$ to be compact. (See Section 4.2 of this paper: The inverse function theorem of Nash and Moser). Notice that as long as the domain $M$ is compact, the questions (1), (2) and (3) become trivial.

**(3)** In general, if $M$ is non-compact, then is $C^\infty(M,N)$ still a Frechet manifold as in the case $M$ is compact?

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