An example of compact diffeomorphism group? Let $X$ be a compact smooth manifold. Then we can endow the set of diffeomorphisms of $X$ with $C^1$ topology. Are there any examples when $\mathrm{Diff}(X)$ is compact (as a topological space)?

It is very far from being compact, its Lie algebra is the set of vector fields which is infinite dimensional if $dim>0$ so it is an infinite dimensional Lie group.

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