Projection of the surface on another surface using Stokes theorem?

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Can multiple surfaces connected together be used to project them onto another surface?

For example let's take a cube. I have seen different examples from books where they project the upper surface and the 4 sides of the cube onto, the bottom surface of the cube. Also, I saw examples where they took the boundary of the bottom surface equal to the boundary of the 5 other surfaces. For example the 5 blue surfaces could be projected onto the red surface.

I know that for example this is possible with a hemisphere were the round upper surface is projected onto the lower surface which is a circle, but could this also apply to other more complex surfaces such that of a cube which is made from multiple surfaces?Image of Cube