First of all, apologies if this question seems "rookie" or shows I lack basic concepts of Differential Geometry. I'm going straight to my question now:
Studying 3+1 Formalism, there was a mapping between a $3D$-manifold and a $4D$-one.
It pushed vectors from $3D$ to $4D$, and pulled -forms from $4D$ to $3D$.
I couldn't understand why you can't reverse map the pull-back and push-forward mapping, and I'm specifically referring to the Remark at the start of page 21 (from page 19.5 starts the whole manifolds thingy) , especially after pull-back mapping definition (one could do somewhat the same for vectors?).
I also saw this answer here on S.E. but this one mainly refers how you can't go from $R^3 \rightarrow R^4$ which was actually done for the vectors via push-forward mappings.
Thank you for your time. Any hint/source will help!