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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 $\mu=\gamma_m|(c,d)$

my question is I confused about the dependency of $a$ , $b$ on $m$ and not getting any feeling of that definition, could any one give one simple example satisfying above conditions?

Next is based on above:

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 that this a map from $M$ to $M$, but not able to visualize what is exactly going on, I need some example on $\mathbb{R}$ or $\mathbb{R}^n$, I will be really happy if some one help me to understand these.

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Simple example: If $X$ is $\mathbb R$, the theorem just states that the ODE $y'=\psi(x)$ has a unique maximal solution through every $(0,y)$ if $\psi$ is $\mathcal C^\infty$. – Henning Makholm Jun 28 '12 at 20:50
thank you~~~~~~~~~ – La Belle Noiseuse Jun 28 '12 at 20:59
up vote 2 down vote accepted

All $(a(m), b(m))$ is doing is defining a "maximal" integral curve starting at the point $m$. The easiest example is if the vector field is complete (e.g. take the manifold to be compact). Then $a(m)=-\infty$ and $b(m)=\infty$ for all $m$. In other words, you can flow along the vector field for all time through any point.

One way to think about this is given some point $m\in M$ you could certainly define some infinitesimally small curve $(-\epsilon, \epsilon)\to M$ through $m$ whose tangent vector field is $X$ along the curve. This will probably extend to some larger domain because this small curve segment is probably a part of some larger integral curve. This theorem tells you that in fact there is a well-defined maximum.

The map $X_t$ is flowing along the vector field for time length $t$. If $t=0$, then $X_0$ is just the identity. You don't flow for any time. If $t=1$, then $X_t(m)$ just means take the point $m$ and move it along the integral curve for a time length of $1$.

In $\mathbb{R}^2$ you could take the constant vector field $\frac{\partial}{\partial x}$. It just points to the right with magnitude $1$ everywhere. Intuitively flowing along this is going to move a point to the right with the proper velocity, and this is true. For example, $X_4((x,y))=(x+4, y)$. The integral curves are defined for all time by $\gamma_{x_0, y_0}(t)=(x_0+t, y_0)$, so $X_t(x,y)=(x+t, y)$.

I think of it like water flowing over the manifold. The vector field tells you the direction and magnitude of the flow of the water at a given point. The integral curve says if you start at some point $m$ it is like dropping something in the water at that point and letting it move with the flow of the water for the length of time specified. The flow $X_t$ is sort of the global picture of doing this with all points simultaneously.

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simply pleased and delighted at your explanation Matt :) – La Belle Noiseuse Jun 28 '12 at 20:58

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