Let us denote the vector fields on as . We can identify each vector field at as , for some real function . So it is clear that . The Lie bracket of two vector fields is then
Diffeomorphism Group of the Circle
Denote the group of orientation preserving diffeomorphisms of the circle as .
The most important property of is that it is a simple group, and it can be generated by the exponential maps.
Lemma
is simple.
Corollary
is generated by the flows of vector fields in the Lie algebra via the exponential map.
Proof First observe that is compact, so every smooth vector field gives rise to a global flow, which is a one-parameter subgroup of diffeomorphisms, denoted as . Since is simple, it suffices to show that is normal in . This follows from the Denjoy’s theorem.
Proposition
is an infinite dimensional Lie group whose Lie algebra is identified with the real topological vector space of smooth vector fields on with the usual topology.
Proof It is clear that forms a group under composition. We first show that is a smooth manifold. First, it is a topological space, because it can be seen as a subspace of with topology. Moreover, the exponential map provides a chart about the identity. Specifically, is a Fréchet space, and consider a map , that sends any function to the corresponding global flow (since is compact, a smooth function in always gives rise to a global flow) generated by . We claim that a neighborhood of , is open in , and