Diffeomorphism group

Diffeomorphism group

Let ${\displaystyle M}$ be a differentiable manifold that is second-countable and Hausdorff. The diffeomorphism group of ${\displaystyle M}$ is the group of all ${\displaystyle C^{r}}$ diffeomorphisms of ${\displaystyle M}$ to itself, denoted by ${\displaystyle {\text{Diff}}^{r}(M)}$ or, when ${\displaystyle r}$ is understood, ${\displaystyle {\text{Diff}}(M)}$. This is a "large" group, in the sense that—provided ${\displaystyle M}$ is not zero-dimensional—it is not locally compact.