Suppose you want to do calculus -- look at issues like smoothness and volume -- in spaces which are not flat. That's what differential topology is all about. Differential geometry uses the tools developed from the calculus to measure things like curvature.

One of the approaches is to describe non-flat, strange spaces as being made up of tiny pieces of regular space that are stitched together. Think of a patchwork quilt placed over somebody who is sleeping. The quilt will have mounds and valleys, creases and corners. But, on the local level (zooming in really close), each little square in the patchwork looks like a flat piece of paper.

Read more abour differential geometry in the Mathematical Atlas.