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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 this question).

• 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:

1. Should the given "$c_*,p$" be $c_*,p: T_p([a,b]) to T_colorredc(p)C$ ?



2. 
   

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

   
   
   
   • $c_*,p: T_p([a,b]) to T_pM$
   
   
   • 
     $j: C to M$ 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.

   
   


3. 
   

   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?
   
   


4. 
   

   What exactly is the orientation on $C$? I think the orientation on $[a,b]$ is given by smooth vector field $fracddx$ on $(a,b)$, smooth outward-pointing vector field $fracddx$ at $x=b$ and smooth outward-pointing vector field $-fracddx$ 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 $c_*,p[fracddxmid_p]$ 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, $c_*,p[fracddxmid_p]$. What's the original orientation supposed to be? We can define the pushforward $c_*[fracddx]$ if $c$ is injective (Subsection 14.5), but how do we know $c$ is injective?
   
   



5. 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)?



6. 
   

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

   
   
   

   • 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 $(2.1)$.

   
   



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

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


2. 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.


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


4. 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 $mathbbR^n$, exactly the derivative of $c$ at $p$, thought of as a vector in the tangent space of $mathbbR^n$ at $c(p)$.


5. 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?


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


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