Terminology "local flow" in Do Carmo's Riemannian geometry

Question

Just trying to understand the use of the term "local flow", and if it has some sort of physical counterpart.

Let $X$ be a differentiable vector field on a differentiable manifold $M$, and let $p \in M$. Then there exist a neighborhood $U \subset M$ of $p$, an interval $(-\delta,\delta)$, $\delta > 0$ and a differentiable mapping $\varphi : (-\delta,\delta) \times U \to M$ such that the curve $t \to \varphi(t,q)$, $t\in (-\delta,\delta)$, $q \in U$, is the unique curve which satisfies $$ \frac{\partial \varphi}{\partial t} = X(\varphi(t,q)) $$ and $\varphi(0,q) = q$. A curve $\alpha : (-\delta,\delta) \to M$ which satisfies $\alpha'(t) = X(\alpha(t))$ and $\alpha(0) = q$ is called a trajectory of the field $X$ that passes through $q$ for $t = 0$. The theorem above guarantees that for each point of a certain neighborhood there passes a unique trajectory of $X$ and that the mapping so obtained depends differentiably on $t$ and on the initial condition $q$. It is common to use the notation $\varphi_t(q) = \varphi(t,q)$ and call $\varphi_t : U \to M$ the local flow of $X$.

Apart from the rigorous discussion of the paragraph above, I was literally trying to understand the name "local flow". Is there a specific meaning behind this terminology? something which is very intuitive maybe.

Answer 1

We can think of the flow in the context of ODEs in $\mathbb{R}^n$. Fix $E$ open, $f\in C^1(E)$. For $x_0\in E,$ let $\varphi(t,x_0)$ denote a solution to $\dot{x}=f(x),\ x(0)=x_0$, defined on a maximal interval of existence. Then, we can define a one-parameter family $\varphi_t$ via $\varphi_t(x_0)=\varphi(t,x_0)$, which we call of the flow of the ODE.

This can be looked at in a few ways. First, we can fix an initial condition, in which case this defines a solution curve through $x_0$. So, for a fixed initial condition, the trajectory can be visualized as motion along a curve through your initial condition. You can also this of the initial condition as varying in a set $E'\subset E$, in which case the map $\varphi_t$ can be visualized as the motion of all of the points in $E'$ (think of a set being "transported"). A nice way to think about this is to view an ODE as describing the motion of a fluid, in which case a trajectory is the motion of a particle in the fluid, and the flow is the motion of the fluid itself.

Answer 2

Let me expand a bit the last sentence of omk's answer, since that's really the reason for the word "flow" in this situation.

Imagine the manifold $M$ as being the space occupied by some liquid. The liquid consists of molecules, but these are so tiny that we usually view the liquid as a continuous entity. Think of the vector field $X$ as telling how the molecules move. More precisely, for any point $m\in M$, a molecule at location $m$ moves with velocity $X(m)$. So $X$ describes the instantaneous motion of the liquid, and we might therefore call it the instantaneous flow.

We might want to describe the motion of the fluid for all times, not just instantaneously. That is, we might want to know where a molecule that is now at position $m$ will be after a time interval $t$. This would be a function $\phi(m,t)$ that could reasonably be called the global flow resulting from the instantaneous flow $X$. There is, however, a problem with this idea of a global flow: Unless we assume some restrictive hypotheses about $X$, the global flow may not exist. An easy but instructive example is to let $M$ be the real line and to define $X(m)=1+m^2$. A molecule that starts at position $0$ will be at position $\tan t$ after time $t$, so it will fall off the end of the manifold $\mathbb R$ at time $\pi/2$. That is $\phi(0,t)$ isn't defined for $t\geq\pi/2$.

What does exist is a little piece of the desired global flow $\phi$, namely $\phi(m,t)$ for $|t|$ smaller than some little $\delta>0$. And this $\delta$ can depend on $m$. What Do Carmo is saying is that, if you restrict $m$ to range over a sufficiently small region region $U\subseteq M$, then you can find a $\delta$ that works for all $m\in U$. So the flow function $\phi$ is well-defined on $U\times(-\delta,+\delta)$. Because it's just a little piece of the desired (but non-existent) global flow, i.e., because it's defined only locally, this $\phi$ is called a local flow.

Notice that there's actually something global about this flow, in that it can be defined near any given point in $M$. But it's definitely local with respect to the time parameter $t$, as the $1+m^2$ example shows, and this locality, measured by $\delta$, cannot in general be taken to be constant over the whole manifold $M$.