Diffeomorphism group

The group acts on points, tensor fields, differential forms, and geometric structures by pushforward or pullback. Common subgroups preserve an orientation, a volume form, a Riemannian metric, or a symplectic form. The quotient by the identity component gives a mapping class group when the chosen topology makes that component meaningful.

For compact $M$, the $C^\infty$ topology gives $\operatorname{Diff}(M)$ the structure of an infinite-dimensional Lie group modeled on smooth vector fields. Noncompact manifolds require more care: compact-open and strong Whitney topologies behave differently, and support conditions may be imposed. These analytic structures are not part of the underlying group definition.

$\operatorname{Diff}(\{\ast\})$ is trivial. Every invertible linear map of $\mathbb R^n$ is a diffeomorphism, so $\mathrm{GL}(n,\mathbb R)$ is a subgroup of $\operatorname{Diff}(\mathbb R^n)$. For the circle, rotations form a subgroup, but the full diffeomorphism group also contains nonlinear reparametrizations.

1. Augustin Banyaga, The Structure of Classical Diffeomorphism Groups, Mathematics and Its Applications 400, Kluwer, 1997. DOI record. Relevant: Chapter 1, diffeomorphism groups and structure-preserving subgroups.
2. Peter W. Michor, Manifolds of Differentiable Mappings, Shiva Mathematics Series 3, Shiva Publishing, 1980. Author-hosted scan. Relevant: differentiable mapping spaces and diffeomorphism groups.