Diffeomorphisms

Local description

Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine ${\displaystyle f}$ going from dimension ${\displaystyle n}$ to dimension ${\displaystyle k}$. If ${\displaystyle n<k}$ then ${\displaystyle Df_{x}}$ could never be surjective, and if ${\displaystyle n>k}$ then ${\displaystyle Df_{x}}$ could never be injective. In both cases, therefore, ${\displaystyle Df_{x}}$ fails to be a bijection.