Diffeomorphism

Definition

Let $U, V \subseteq \mathbb{R}^n$ be open, and let $k \ge 1$ be an integer or $k = \infty$. A map $f \colon U \to V$ is a $C^k$ diffeomorphism if $f$ is a bijection of class C^k on $U$. In addition, $f^{-1} \colon V \to U$ must be of class $C^k$ on $V$. The sets $U$ and $V$ are $C^k$ diffeomorphic if there is a $C^k$ diffeomorphism from $U$ to $V$.

Remarks

If $f$ is a diffeomorphism, the chain rule applied to $f^{-1} \circ f = \mathrm{id}$ shows that $Df(x)$ is invertible for each $x \in U$. Conversely, a $C^k$ bijection $f \colon U \to V$ with $Df(x)$ invertible at each point is a $C^k$ diffeomorphism, by the inverse function theorem. The map $x \mapsto x^3$ is a $C^\infty$ bijection of $\mathbb{R}$, but it is not a diffeomorphism, because its inverse $x \mapsto x^{1/3}$ is not differentiable at $0$. Polar coordinates $(r, \theta) \mapsto (r \cos \theta, r \sin \theta)$ give a $C^\infty$ diffeomorphism from $(0, \infty) \times (0, 2\pi)$ onto the plane without the closed half-line $\{(x, 0) : x \ge 0\}$.

Explanations

L1 human

A diffeomorphism is a smooth change of coordinates that can be undone smoothly. It may stretch and bend space, but it does not tear, fold or crush it.