In $M^4=S^1 \times RP^3$ with a real projective space $RP^3$, could we construct a Möbius band M$^2$ which is a nontrivial line bundle over $S^1$ such that:
- this Möbius band M$^2$ is embedded in $M^4=S^1 \times RP^3$, and
- this Möbius band M$^2$ shares the same $S^1$ of $M^4$?
If yes, the Möbius band M$^2$ can be embedded in $M^4=S^1 \times RP^3$, then it seems strange because the $M^4$ is a trivial bundle $RP^3$ over $S^1$. Could you explain why this still a yes construction?
If no, could we find another 4-manifold $N^4$ (also with a $S^1$ circle) such that the Möbius band M$^2$ is embedded in $N^4$? For example, should $N^4 = S^1 \ltimes RP^3$ and M$^2 = S^1 \ltimes I^1$? (what exactly is this fibration construction?)