When you avoid tight bends and creases (singularities---places where the derivative vanishes), you have a nice structure on which you can do useful things like

When you avoid tight bends and creases (singularities---places where the derivative vanishes), you have a nice structure on which you can do useful things like take derivatives, measure lengths and angles, find "straight" lines, and such. In short, the structure behaves, as long as you're small enough, just like the space we live in and have studied for centuries. Now you can extend this notion from a property of one surface to a property of a continuous family of surfaces, and then look at the beginning and the end (imagine it varying with respect to time, just like in the video). The niceness property above, if maintained throughout, gives you more things to say about what happens to substructures. For example, you can say something like "if I start with a triangle inside this surface, and morph it into another surface keeping it nice and smooth the whole time, I'll still get a triangle, and it won't do {big class of bad things}".