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My book is An Introduction to Manifolds by Loring W. Tu. The following is an entire subsection (Subsection 22.5) of the section that introduces manifolds with boundary (Section 22, Manifolds with Boundary).

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Note: I believe that all manifolds with or without boundary referred in this subsection have unique dimensions by some convention (either it's implicit, or it's explicit a I missed it) in the section, contrary to the convention of the book (See here and here).

According to an errata by Ehssan Khanmohammadi, the only erratum to be made in this subsection is that $c((0,\varepsilon[) \subset M^\circ$ should be changed to $c(]0,\varepsilon[) \subset M^\circ$. I still have several concerns about this subsection.

  1. What exactly is a vector field along $\partial M$, and what is its domain?

  2. For the local expression of $X$, a vector field along $\partial M$ is the following understanding correct?

    • 2.1 Let $p$ be in the domain of $X$, which is $\partial M$ (see question 1). View $p$ as an element of $M$, which we can do because $\partial M \subseteq M$. We view $p$ as such to obtain a coordinate neighborhood $(U,\varphi) = (U,x^1, ..., x^n)$ of $p$ in $M$ to obtain a local expression for $X$ in $(U, \varphi)$ specifically in $(U \cap \partial M, \varphi|_{U \cap \partial M})$.

    • 2.2 In the expression $$X_q = \sum_{i=1}^n a_i(q) \frac{\partial}{\partial x^i}|_q, q \in \partial M,$$

      • 2.2.1. this is a local expression so "$q \in \partial M$" means "$q \in \partial M \cap U_p$", like in the proof that $\partial M$ is a manifold (without boundary).

      • 2.2.2. the $a_i$'s are $a_i: U \cap \partial M \to \mathbb R$, functions on $U \cap \partial M$ rather than functions on $U$, but I'm not quite sure of this based on the "$a^i$ on $\partial M$" in the smoothness (see question 3)

      • 2.2.3. the "$x_i$"'s are actually the $x_i|_{U \cap \partial M}$'s from $(U \cap \partial M, \varphi|_{U \cap \partial M}) = (U \cap \partial M, x^1|_{U \cap \partial M}, ..., x^n|_{U \cap \partial M})$, i.e. they are restrictions of the original $x_i$'s rather than the original $x_i$'s themselves, though there is a convention in this book to omit indicating restrictions (see here, here and here).

  3. Is this a correct understanding of the smoothness definition?

Let $M$ be a manifold with boundary $\partial M$. Let $X$ be a vector field along $\partial M$. For each $p$ in the domain of $X$, which is $\partial M$ (see question 1), $X$ has a local expression at $p$: For any coordinate neighborhood $(U, \varphi) = (U, x^1, ..., x^n)$ of $p$ in $M$, we have a coordinate neighborhood of $p$ in $\partial M$ namely the restriction of $(U, \varphi) = (U, x^1, ..., x^n)$ which is $(U \cap \partial M, \varphi_{U \cap \partial M}) = (U \cap \partial M, x^1|_{U \cap \partial M}, ..., x^n|_{U \cap \partial M})$, and hence the local expression is as follows

$$X_q = \sum_{i=1}^n a_i(q) \frac{\partial}{\partial x^i}|_q, q \in \partial M \cap U \tag{1}$$

We define $X$ to be smooth at $p$, if for all coordinate neighborhoods $(U, \varphi) = (U, x^1, ..., x^n)$ of $p$ in $M$, we have that for the (I say "the" because I guess the restriction $(U \cap \partial M, \varphi_{U \cap \partial M})$ is unique for a given $(U, \varphi)$) corresponding coordinate neighborhood $(U \cap \partial M, \varphi_{U \cap \partial M}) = (U \cap \partial M, x^1|_{U \cap \partial M}, ..., x^n|_{U \cap \partial M})$ of $p$ in $\partial M$ that gives the local expression for $X$ as in $(1)$ that there exists a neighborhood $W$ of $p$ in $U$ (I'm going to ignore the details of "neighborhood" as in open subset versus "coordinate neighborhood" as in open subset along with homeomorphism and treat "neighborhood" and "coordinate neighborhood" equivalently and thus omit assigning some $\varphi$ or $\psi$ or whatever to $W$) such that the functions $a_i|_W$ are smooth at p.

  • Now, such neighborhood $W$ of $p$ in $U$ is also a neighborhood of $p$ (in $M$) because $W$ is open in $M$ because $U$ is open in $M$.

  • I guess I take the "$a^i$ on $\partial M$" to mean "$a^i$ on $\partial M \cap W$". I could be missing something that actually suggests that the $a_i$ are originally on $U$ or $M$ or something and then "$a^i$ on $\partial M$" means, respectively, $a_i|_{U \cap \partial M}$ or $a_i|_{(M \cap) \partial M}$.

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  1. Despite the title of the subsection, I don't think there's a definition for outward-pointing vector fields. What is it exactly?

  2. In the proof of Proposition 22.10, is it understood that we cover $\partial M$ by restrictions of the $(U_{\alpha}, x^1_{\alpha}, ..., x^n_{\alpha})$'s like in questions 2 and 3?

  3. Actually, based on Lee's Problem 8-4, asked about here, I think we can interpret Proposition 22.10 without the concept of "along" as follows:

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  • $\begingroup$ Did you happen to find an answer to your question? If not, I would recommend cleaning it a bit, leaving only the essential. It makes it more appealing to the answer giver. $\endgroup$ – An old man in the sea. Jun 12 at 18:38

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