Definition 9.2:
Let
be a set. A flow on
is a group action whose group is
.
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
![{\displaystyle \Phi _{\mathbf {V} }:\{(x,p)|p\in M,x\in I_{p}\}\to M,\Phi _{\mathbf {V} }(x,q):=\gamma _{q}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/858205d0f096b41c542eaaf9483c47f8f6f72934)
Further, for all
, we define the function
![{\displaystyle \Phi _{\mathbf {V} ,x}:\{p\in M|x\in I_{p}\}\to M,\Phi _{\mathbf {V} ,x}(q):=\Phi _{\mathbf {V} }(x,q)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a285cdf4c66776134ac3e89b36e10fe083e2497a)
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
![{\displaystyle \rho _{p}:\mathbb {R} \to M,\rho _{p}(x):=\gamma _{p}(y+x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e94cf5e28196ce9baf3f1bd8ea2b0c99373b1068)
, then using the fact that
is an integral curve of
, we obtain for all
, that
![{\displaystyle (\varphi \circ \rho _{p})'(x)=(\varphi \circ \gamma _{p})'(x+y)=(\gamma _{p})_{x+y}'(\varphi )=\mathbf {V} (\gamma _{p}(x+y))(\varphi )=\mathbf {V} (\rho _{p}(x))(\varphi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61337ffb679cc885755edffb84e3fe48a4d057e8)
Hence, since
and
are both integral curves and furthermore
![{\displaystyle \rho _{p}(0)=\gamma _{p}(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa00359f28f78220e9110d9f056b914a72459a63)
due to theorem 8.2 follows
and therefore
![{\displaystyle {\begin{aligned}\Phi _{\mathbf {V} }(x,\Phi _{\mathbf {V} }(y,p))&=\Phi _{\mathbf {V} }(x,\gamma _{p}(y))\\&=\gamma _{\gamma _{p}(y)}(x)\\&=\rho _{p}(x)\\&=\gamma _{p}(x+y)\\&=\Phi _{\mathbf {V} }(x+y,p)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/937ce62ff6acc75e65e1a7a33f3ddad3268e6724)
2. Since
is the identity element of the group
, we have
![{\displaystyle \Phi _{\mathbf {V} }(e,p)=\Phi _{\mathbf {V} }(0,p)=\gamma _{p}(0)=p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5b5baaffb533de5520719c66e611a410c13525c)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
Proof:
Let
be arbitrary. We have:
![{\displaystyle {\begin{aligned}\lim _{h\to 0}{\frac {\Phi _{\mathbf {V} ,h}^{*}(\varphi )(p)-\varphi (p)}{h}}&=\lim _{h\to 0}{\frac {\varphi (\Phi _{\mathbf {V} ,h}(p))-\varphi (p)}{h}}\\&=\lim _{h\to 0}{\frac {\varphi (\gamma _{p}(h))-\varphi (p)}{h}}\\&=(\gamma _{p})_{0}'(\varphi )\\&=\mathbf {V} (p)(\varphi )=:{\mathfrak {L}}_{\mathbf {V} }\varphi (p)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bda873c208c2f2e12aecdf06ac15c90c6bb9c68c)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
Corollary 9.6:
From the definition of
, we obtain:
![{\displaystyle \forall p\in M:\mathbf {V} \varphi (p)=\lim _{h\to 0}{\frac {\Phi _{\mathbf {V} ,h}^{*}(\varphi )(p)-\varphi (p)}{h}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81de1e28987eb7c263504fc3b82e7316f8e21340)
Proof:
Let
and
be arbitrary. Then we have:
![{\displaystyle {\begin{aligned}\lim _{h\to 0}{\frac {\mathbf {W} (\Phi _{\mathbf {V} ,h}(p))\circ \left(\Phi _{\mathbf {V} ,h}^{-1}\right)^{*}(\varphi )-\mathbf {W} (p)(\varphi )}{h}}&=\lim _{h\to 0}{\frac {\mathbf {W} (\Phi _{\mathbf {V} ,h}(p))(\varphi \circ \Phi _{\mathbf {V} ,h}^{-1})-\mathbf {W} (p)(\varphi )}{h}}\\&=\lim _{h\to 0}{\frac {\mathbf {W} (\Phi _{\mathbf {V} ,h}(p))(\varphi \circ \Phi _{\mathbf {V} ,h}^{-1})-\mathbf {W} (p)(\varphi )}{h}}+\mathbf {W} (p)(\mathbf {V} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1665e4683ea7dca2f7e9d8bd6ebec4ea73e3d74e)
![{\displaystyle \left|\mathbf {W} (p)(\varphi )-\mathbf {W} \left(\Phi _{\mathbf {V} ,h}^{-1}(p)\right)\left({\frac {\varphi \circ \Phi _{\mathbf {V} ,h}-\varphi }{h}}\right)\right|\leq \left|\mathbf {W} (p)(\varphi )-\mathbf {W} (p)\left({\frac {\varphi \circ \Phi _{\mathbf {V} ,h}-\varphi }{h}}\right)\right|+\left|\mathbf {W} (p)\left({\frac {\varphi \circ \Phi _{\mathbf {V} ,h}-\varphi }{h}}\right)-\mathbf {W} \left(\Phi _{\mathbf {V} ,h}^{-1}(p)\right)\left({\frac {\varphi \circ \Phi _{\mathbf {V} ,h}-\varphi }{h}}\right)\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0f03dcc21f7981517799444a0bd45e8e2574c4)
Let
be contained in the atlas of
such that
. We write
![{\displaystyle \mathbf {W} (q)=\sum _{j=1}^{d}\mathbf {W} _{\phi ,j}(q)\left({\frac {\partial }{\partial \phi _{j}}}\right)_{q}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4aaed85d99b6fe28837cc7dcd81670e4a93baa1)
for all
.
We now choose
such that
(which is possible since
is open as
is in the atlas of
). If we choose
we have
![{\displaystyle \mathbf {W} \left(\Phi _{\mathbf {V} ,h}^{-1}(p)\right)=\sum _{j=1}^{d}\mathbf {W} _{\phi ,j}(\Phi _{\mathbf {V} ,h}^{-1}(p))\left({\frac {\partial }{\partial \phi _{j}}}\right)_{\Phi _{\mathbf {V} ,h}^{-1}(p)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4564733f0ae57a5261eaf5f26562a55cb8d16181)
From theorem 5.5, we obtain that all the functions
are contained in
.
Corollary 9.8: