What is the induced orientation on a 1-manifold with boundary that is the image of closed interval under a smooth immersion? My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary.

I have been trying to wrap my head around this for about 2 hours (3.5 hours, if you include the 1.5 hours spent on typing up this question).


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*The context of this example is the preceding example and Example 22.9 which are examples of the preceding Propositions 22.11 and 22.12,

*I guess we use positive linear maps to create analogous atlases for $[a,b]$ from the atlases for $[0,1]$ (one of them was an oriented atlas and the other wasn't), so I get why $[a,b]$ is a smooth oriented manifold with boundary, but what I don't get is almost everything after "An orientation on $[a,b]$".

*I am trying to not use the classification of smooth 1-manifolds with boundary (since such classification is not so far given in this book, although I discovered such classification from another book, Introduction to Smooth Manifolds by John M. Lee (Jack Lee)):
My questions are:


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*Should the given "$c_{*,p}$" be $c_{*,p}: T_p([a,b]) \to T_{\color{red}{c(p)}}C$ ?

*Is the given "$c_{*,p}: T_p([a,b]) \to \{\text{see (1) for range}\}$" actually $(j \circ c)_{*,p} = j_{*,c(p)} \circ c_{*,p}$ where


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*$c_{*,p}: T_p([a,b]) \to T_{c(p)}M$

*$j: C \to M$ and $j_{*,c(p)}: T_{c(p)}C \to T_{c(p)}M$, both are inclusion,

*so the given "$c_{*,p}$" is an "induced" differential, where "induced" refers to restricting range like in Subsection 11.4 ?

*2.1. Is the given "$c_{*,p}$" then an isomorphism and thus $c$ is a local diffeomorphism by Remark 8.12 on the Inverse Function Theorem? How is this relevant? I think this answers question (6) below.


*It's not stated as to what $M$ is, but I think $M$ is a smooth oriented n-manifold with boundary. Is this relevant, and why or why not?


*

*3.1. Must $n=1$ in this example?


*What exactly is the orientation on $C$? I think the orientation on $[a,b]$ is given by smooth vector field $\frac{d}{dx}$ on $(a,b)$, smooth outward-pointing vector field $\frac{d}{dx}$ at $x=b$ and smooth outward-pointing vector field $-\frac{d}{dx}$ at $x=a$ and orientation form $dx$ on all of $[a,b]$ (I think it's the same form for each boundary point and for the interior unlike with the vector field), so for $C$, I think the smooth outward-pointing vector field is $c_{*,p}[\frac{d}{dx}\mid_p]$ and something to do with $c$ and $dx$ like $c^{*}(dx)$, $d(c \circ x)$ or $c \circ (dx)$.


*

*4.1. Also I seem to have only a local orientation at $p$, namely, $c_{*,p}[\frac{d}{dx}\mid_p]$. What's the original orientation supposed to be? We can define the pushforward $c_{*}[\frac{d}{dx}]$ if $c$ is injective (Subsection 14.5), but how do we know $c$ is injective?


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*4.1.1. There might be other ways to define the pushforward. Hopefully at least one of those pushforward definitions is smooth. I'm about to read more here.



*Where do we use injectivity of $c_{*,p}$, either the original or the given "$c_{*,p}$" (whose injectivity follows from composition of injections is an injection)?

*How do we know $\partial (c[a,b]) = c (\partial [a,b])$ and $ (c[a,b])^o = c ([a,b]^o)$?


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*I think this would follow from Proposition 22.4 if $c$ were injective, but (see question $(4.1)$).

*I think this would follow from Proposition 22.4 if $c$ were a local diffeomorphism, which I think follows from a "yes" to question $(2.1)$ or if $c$ were a local diffeomorphism onto its image (which I think means that the restricted range $c$, $c: [a,b] \to c[a,b]$ is a local diffeomorphism)


*Are "sections" relevant? I think even if $c$ is not injective, $c$ can have sections even if $c$ has no inverse or something.
 A: *

*Yes, we have $p \in [a,b]$, not in $M$.

*I think that somehow immersed submanifolds should be included under consideration by this example, otherwise the wording is needlessly cumbersome. In this case I think your intuition is right, but If $M$ has dimension two, say, $c$ should not be a local diffeomorphism. Well, perhaps onto its image, but that's exactly what an immersion is already.

*In this example we should consider $M$ that have orientation and those that do not, boundary or not, and the dimension should be arbitrary.

*The orientation on the image of $C$ is more or less as you describe. Think of the vector field version of it as the tangent line to the curve, pointing in the direction of the orientation on $[a,b]$, i.e. if $M$ is $\mathbb{R}^n$, exactly the derivative of $c$ at $p$, thought of as a vector in the tangent space of $\mathbb{R}^n$ at $c(p)$.

*Actually here I think all we use is that $c$ is a smooth function. The immersion condition also probably tells us that at any points $p$ and $q$ with $c(p) = c(q)$, the tangent vectors point in different directions?

*Here I think being locally injective should be good enough? Take some small neighborhood and see if 22.4 gets you there?

*I don't think so for this example? I don't think about differential geometry every day, so I could be wrong.

