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Warner page-37: Theorem: Let $X$ be a $C^{\infty}$ vector field on a differentiable manifold $M$ For each $m\in M$ there exist $a(m)$ and $b(m)$ in extended real line, and smooth curve $$\gamma_m:(a(m),b(m))\rightarrow M$$ such that

a) $0\in (a(m),b(m))$ and $\gamma_m(0)=m$

b)$\gamma_m$ is an integral curve of $X$

c) If $\mu:(c,d)\rightarrow M$ is a smooth curve satisfying conditions $a$ and $b$ then $(c,d)\subseteq (a(m),b(m))$ and $\mu=\gamma_m|(c,d)$

Next is:

Definition: $\forall t\in \mathbb{R}$ we define a transformation $X_t$ with domain $$D_t=\{m\in M:t\in (a(m),b(m))\}$$ by setting $$X_t(m)=\gamma_m(t),$$ I understand the transformation but not able to prove any one of the following:

d) For each $m\in M$ there exists an open neighborhood $V$ of $m$ and $\epsilon>0$ such that the map $$(t,p)\mapsto X_t(p)$$ is defined and is $C^{\infty}$ from $(-\epsilon,\epsilon)\times V\rightarrow M$

e) $D_t$ is open.

f) $\bigcup_{t>0} D_t=M$

g)$X_t:D_t\rightarrow D_{-t}$ is a diffeomorphism

h) let $s$ and $t$ be the real numbers. Then the domain of $X_s\circ X_t$ is contained in but generally equal to $D_{s+t}$. However the domain of $X_s\circ X_t$ is $D_{s+t}$ in the case in which both $s$, $t$ have the same sign. More over on the domain of $X_s\circ X_t$ we have $$X_s\circ X_t=X_{s+t}$$

I must confess that I am really not getting clearly the proofs written the book after reading so many times last 2 days :(, could any one help me to write and learn this things formally and rigorously step by step and if possible with some informal pictures :(. I really need to understand this.please help. thank you.

I understand about the transfromation $X_t$ and imagine like that: first of all it is defined on some subset of $M$, suppose I take a point $m\in M$ and for that $m$ I am getting $a(m), b(m)$ I mean $(a(m),b(m))$ which is open interval, I imagine this line is passing perpendicularly through the point $m$, so informally for each $m\in M$ I am getting things like $m\times (a(m),b(m))$ and so if I vary my $m$ I can imagine a cylinder whose top may not be a cicrle as $a(m),b(m)$ depends on $m$. and my $t\in (a(m),b(m)$, and My map is from those $m\in M$ as we defined $D_t$ to $M$ just $X_t(m)= \gamma_m(t)$ I mean the point on the curve when the curve passes at time $t$

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I think it would be better to use a different character for the map $X_t$, e.g. $\phi_t$ just to avoid confusion. – Mercy King Jul 1 '12 at 16:31

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