Does classification of 1-manifolds with boundary give induced orientation of image of closed interval under a smooth immersion?

geometry differential-geometry general-topology derivatives manifolds

You are right that the classification does not imply that $C$ and $[a, b]$ are diffeomorphic. It may very well be that $C$ is diffeomorphic to a circle, and that $c$ winds the interval $[a, b]$ 3.5 times around it. (The minute hand of the clock was giving you a hint.)

But assuming that $C$ has nonempty boundary, it follows indeed that the immersion $c$ is injective (i.e. is an embedding), as shown in a linked question: Is a smooth immersion $c: [a,b] \to M$ injective if its image is a 1-manifold with non-empty boundary?

In particular, the boundary of $C$ has cardinality $2$ and equals $\{ c(a), c(b) \}$.

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