Pullback (differential geometry)

Hey kiddo, have you ever played with a slingshot? When you pull the rubber band back, you create tension and store energy. Then when you let go, the rubber band snaps forward, releasing that energy and propelling the object you're aiming at.

Now imagine that instead of a slingshot, you have a curved track that a ball can roll along. If you pull the ball back a little bit, it will still be moving toward the end of the track, but at a slower pace. This means that it will take longer to travel the same distance as if it had started from the beginning of the track.

In math, we call this slowing down of movement a "pullback." It's a concept used in differential geometry, which is a fancy way of talking about how shapes change when you bend or stretch them in different ways.

For example, let's say you have a function that maps points in one space (let's call it space A) to points in another space (space B). The pullback of this function would be a way of mapping points in space B back to space A, but with certain adjustments that take into account how the points move along the curve or surface of space B.

This might all sound a bit complicated, but the basic idea is that pullbacks help us understand how shapes and spaces are related to each other. It's kind of like taking a step back to see the big picture, so that we can better appreciate the details of how things move and change over time.