Definition:Diffeomorphism

Open sets in $\R^n$
Let $n$ and $k$ be natural numbers.

Let $U,V\subset \R^n$ be open sets.

Let $f : U \to V$ be a mapping.

Then $f$ is a $C^k$-diffeomorphism $f$ is a bijection of class $C^k$ with an inverse of class $C^k$.

Differentiable Manifolds
Let $m,n\geq0$ and $k$ be natural numbers with $1\leq k \leq \min(m,n)$.

Let $M$ and $N$ be differentiable manifolds of dimensions $m$ and $n$.

Let $f : M \to N$ be a mapping.

Then $f$ is a $C^k$-diffeomorphism $f$ is a bijection of class $C^k$ with an inverse of class $C^k$.

Smooth Diffeomorphism
A smooth diffeomorphism is a bijection which is smooth and whose inverse is smooth.