Differentiable Manifolds/Group actions and flows

Integral curves and Lie derivatives Product manifolds and Lie groups Definitions of group actions and flows 0 100% #FFFFFF Definition 9.1: Let be a set, be a group, and the identity of . A group action is a function such that for all and all : 0 100% #FFFFFF Definition 9.2: Let be a set. A flow on is a group action whose group is . The flow of a vector field 0 100% #FFFFFF Definition 9.3: Let be a manifold of class , where ( must be here), and let . Due to theorem 8.2, for each exists a maximal open interval such that and such that there is a unique curve such that and is an integral curve of . Then the flow of is defined as the function Further, for all , we define the function 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. Proof: Let be arbitrary. 1. 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 0 100% #FFFFFF Theorem 9.5: Let be a manifold of class , let , let be the flow of , and let be arbitrary. Then we have: Proof: Let be arbitrary. We have: Corollary 9.6: From the definition of , we obtain: 0 100% #FFFFFF Theorem 9.7: Let be a manifold of class and let be vector fields. Then we have: Proof: 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 . Corollary 9.8: Integral curves and Lie derivatives Product manifolds and Lie groups