I am going over a paper in which a theorem includes a condition that certain arcs are "properly-immersed geodesic arcs" in a surface $\Sigma$ with nonempty boundary. I have done a search, but I have not been able to find a definition of this term. I understand the individual parts, but not the whole; I understand that a properly-embedded arc $\alpha : [0,1] \rightarrow \Sigma$ in a surface $\Sigma$, is one whose endpoints are in $\partial \Sigma $. I assume an immersed arc $\alpha$ is one whose image $\alpha ([0,1]) \subset \Sigma$ is an immersed submanifold of $\Sigma$.
Is this all there is to the concept of a properly-immersed arc $\alpha$ , i.e., $\partial \alpha \subset \partial \Sigma$ and the image $\alpha([0,1])$ is an immersed submanifold of $\Sigma$ ?
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$\begingroup$An map of manifold pairs $$ f: (N, \partial N)\to (M, \partial M) $$ is called proper if $f^{-1}(\partial M)=\partial N$. Here I am assuming that $N$ is compact. Now, apply this to maps which are immersions and $N$ which is a closed interval.
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