Motivation: If you have a cube, then you know that there is a planar figure, consisting of 6 squares, such that if you glue carefully, then you got your cube. Then, can we figure out that whether a given paper (with a specific shape) form a closed surface without boundary or not?
Example: A square (or rectangle) definitely form a torus, if the size of square is suitably big so that you don't have any problems to glue them as in typical topology course.
Counterexample: However, a trapezoid cannot form a closed surface without boundary, even if it may form a Mobius strip. It was wrong as in the comment. Thank Rob!
Question: Is there a theory or theorems telling us that whether a given simply connected surface with boundary form a closed surface without boundary only using isometries over the boundaries (not continuous deformation, which was used in topology)? Any recommendation for references are welcome!