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.)

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