Differentiable Manifolds/Group actions and flows
Definitions of group actions and flows edit
The flow of a vector field edit
Theorem 9.4: Let be a manifold of class , where ( must be ), let and let be the flow of . If for each the interval such that there is a unique curve such that and is an integral curve of is equal to , then the flow of is a flow.
Let be arbitrary.
If we choose in the atlas of such that and further define
, then using the fact that is an integral curve of , we obtain for all , that
Hence, since and are both integral curves and furthermore
due to theorem 8.2 follows and therefore
2. Since is the identity element of the group , we have
Let be arbitrary. We have:
From the definition of , we obtain:
Let and be arbitrary. Then we have:
Let be contained in the atlas of such that . We write
for all .
We now choose such that (which is possible since is open as is in the atlas of ). If we choose we have
From theorem 5.5, we obtain that all the functions are contained in .