Differentiable Manifolds/Group actions and flows

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Definitions of group actions and flows

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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  :

  •  
  •  

Definition 9.2:

Let   be a set. A flow on   is a group action whose group is  .

The flow of a vector field

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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

  

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:

 

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:

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