Differentiable Manifolds/Integral curves and Lie derivatives

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Integral curves

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Definition 8.1:

Let   be a manifold, let  , and let   be an interval. An integral curve for   is a function   such that

 

Theorem 8.2:

Let   be a manifold of class  ,   (it is important that   here), let   be differentiable of class  ,   and let  . Then there exists an interval   and an integral curve   of   such that   and  .

Proof:

Let   be arbitrary, and let   be contained in the atlas of   such that  .

Lemma 2.3 stated that if we denote  , then the  ,   are contained in  . From   being differentiable of class   with a  , it follows that the functions  ,   are contained in  .

Thus the Picard–Lindelöf theorem is applicable, and it tells us, that each of the initial value problems

 ,  
 

has a solution  , where each   is an interval containing zero. We now choose

 

and

 

We note that

 

Therefore we have for each   and  :

 

Because of theorem 2.7 then follows:

  

Lie derivatives

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In the following, we will define so-called Lie derivatives, for

  •   functions and
  • for vector fields.

Definition 8.3:

Let   be a manifold of class  ,   and  . The Lie derivative of   in the direction of  , denoted by  , is defined as follows:

 

Definition 8.4:

Let   be a manifold and  . The Lie derivative of   in the direction of  , denoted by  , is defined as follows:

 

So we simply defined the Lie derivative of a function in the direction of a vector field as the function defined like in definition 5.1, and the Lie derivative of a vector field in the direction of the other vector field as the lie bracket of first the first vector field and then the other (the order is important here because the Lie braket is anti-symmetric (see theorem ? and definition ?)). Since we already had symbols for these, why have we defined new symbols? The reason is that in certain circumstances, the Lie derivatives really are derivatives in the sense of limits of differential quotients, as is explained in the next chapter.

Differentiable Manifolds
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