"Immediate" Applications of Differential Geometry

When wrapping a ball as a birthday or Christmas present, one cannot avoid the need to crease the paper. This is due to the paper having zero Gaussian curvature, and the ball having positive Gaussian curvature. (Theorema Egregium)

This one is especially appropriate for a class project: It's well known that if you're eating a slice of pizza and the front end sags, you're supposed to fold it lengthwise (so that the crust gets folded against itself). The reason for this is much less well known: pizza has Gaussian curvature 0, so if you create curvature in the left-right direction, it will be forced to remain straight in the lengthwise direction!

(I'm pretty sure I stole this from an old MathOverflow question.)

Differential geometry explains why your telephone cord gets knotted. Most people pick up the telephone receiver with one of their hands- WLOG with the right hand. So when they pick it up and put it down, they make a clockwise motion. This creates writhe. Writhe plus twist gives linking (this is sometimes known as Călugăreanu's Theorem), and the telephone chords gets supercoiled, and can become knotted because linking relaxes to loops, and anything passing through those loops will create trefoil knots.

Indeed, the above process, to the best of my knowledge, outlines how all knotting occurs in nature. Naturally occurring knots have high writhe- lots of trefoils and torus knots, very few Figure Eight Knots.