diffeomorph

A diffeomorphism is a type of smooth map between smooth manifolds that has a smooth inverse. In simpler terms, it's a function that smoothly transforms one space into another, and you can smoothly transform it back. Both the function itself and its inverse must be differentiable at every point, and their derivatives must also be continuous. This means that the transformation doesn't create any sharp corners, breaks, or other irregularities in the structure of the manifolds.

The concept of a diffeomorphism is fundamental in differential geometry and topology. It provides a precise way to define when two smooth manifolds are "the same" in a smooth sense, meaning they are diffeomorphic. If two manifolds are diffeomorphic, they share many essential geometric properties. For example, if one manifold is curved, the other will also be curved in a corresponding way. Diffeomorphisms are the isomorphisms in the category of smooth manifolds.

Examples of diffeomorphisms include linear transformations of Euclidean space that are non-singular, and smooth deformations of geometric shapes that preserve their smoothness. A simple example in 2D would be smoothly stretching a circle into an ellipse. The stretching function and its inverse (shrinking) are both smooth. However, mapping a line segment onto a circle is not a diffeomorphism because the inverse function would not be smooth at the point where the line segment is compressed to a single point.