Boundary parallel

Consider the annulus ${\displaystyle I\times S^{1}}$. Let π denote the projection map

${\displaystyle \pi \colon I\times S^{1}\rightarrow S^{1},\quad (x,z)\mapsto z.}$

If a circle S is embedded into the annulus so that π restricted to S is a bijection, then S is boundary parallel. (The converse is not true.)

If, on the other hand, a circle S is embedded into the annulus so that π restricted to S is not surjective, then S is not boundary parallel. (Again, the converse is not true.)

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  An example wherein π is not bijective on S, but S is ∂-parallel anyway.
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  An example wherein π is bijective on S.
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  An example wherein π is not surjective on S.