Attached, a screenshoot from the book "The geometry of physics" by Theodore Frankel
I would like to better understand what the theorem means.
- First question : Did they forgot in the theorem to say that $\Phi(0,q)=q$ ?
- Second question : Here we say it is possible to find a flow starting in a neighborhood of $p$ where the vector field represent the velocity of this flow. But why only on a neighborhood ? As we work in $\mathbb{R}^n$ and the vector field is $C^k$ in $U$, could'nt we start a flow anywhere in $U$ ? If would like a visual example illustrating that what I say is not true in general if possible.
- Third question : I am not sure to understand the english of the sentence "Any two such curves are equal on the intersection of their t-domaines ("uniqueness")" in the second paragraph. Do they mean that if two curves cut then they are identical (because of the first order differential equation).
- Last question : I am not sure to understand the last paragraph (maybe because of english or because of my math comprehension I don't know). Do they mean the following thing : the ensemble of curves $\{ \phi_t\}$ is not a group because with $\epsilon=1/2$ : $\phi_{1/2}$ belongs to the ensemble but $\phi_{1/2} \circ \phi_{1/2}$ not (the law is not "closed"). In fact that is the final sentence that I don't understand "the point is that $\phi_{1/2}(q)$ need not be in the set $U_p$ on which $\phi_{1/2}$ is defined".