Consider a curve $\gamma: [a,b] \rightarrow M$ where $M$ is some manifold and let $X$ be a vector field. The curve $\gamma$ is said to be an integral curve starting at $p$ if
$$\gamma'(t) = X_{\gamma(t)}, \quad \gamma(0) = p. $$
Given a smooth vector field $X$ and some $p \in M$, we know from ODEs that an integral curve corresponding to $X$ and starting at $p$ always exists and has a unique smooth solution defined on some interval $[a(p), b(p)]$.
In contrast, a local flow is defined about a point $p$ is defined as a smooth map
$$F: (-\epsilon, \epsilon) \times V \rightarrow U$$
where $\epsilon > 0$ and $p \in V \subset U \subset M$, and $F$ is also a one-parameter group.
I am trying to visualize what the difference is between these two concepts other than a flow being a collection of curves (one for each point in $V$) that varies continuously from one initial point to another, as opposed to just one curve. For example, if we consider a particle in some fluid starting at $p$, the integral curve gives the position of the particle at time $t$. It seems to me that a local flow tells us the exact same information, only it also tells us the path if we were to start at some other point $q \in V$. Is this extra information of smooth dependence on initial conditions useful?
Furthermore, the one parameter group is always defined at a specific point $p \in M$, so why don't we define a flow starting from the definition of an integral curve (that starts at some point $p$) as opposed to a function of two variables? In other words why don't we define a flow as: Suppose $\gamma$ is an integral curve defined on $[a, b]$ and starting at $p$. If $t_1$ and $t_2$ are in $[a, b]$ such that $t_1 + t_2 \in [a,b]$, a flow is an integral curve such that $\gamma_{t_1}(p) \circ \gamma_{t_2}(p) = \gamma_{t_1 + t_2}(p)$?
Best Answer
It might be good to keep in mind the earlier discussion Understanding the relationship between local flows and vector fields.
It is. Integral curves are by definition objects attached to points, whereas a local flow is a global object attached to the whole manifold. Thinking of it as a time evolution, one could indeed theoretically consider a single point $p$ and track how it moves under the time evolution, but for practical purposes (as well as for contexts where it is not very reasonable to keep track of individual points; e.g. systems with high degree of freedom like gas in a chamber) it is useful to be able to keep track of the time evolutions of an aggregate of points. One may further need this aggregate of points itself to have a compatible differentiable structure (so that instead of an amorphous set we have an embedded submanifold, like an open ball w/r/t some metric), to be able to linearly approximate the data. But then the differentiable structure of the aggregate of points would need to be preserved under the time evolution for the linear approximations to not break down.
As I had mentioned in the aforementioned discussion indeed one can consider only one trajectory. On the other hand like you say one can consider a (local) flow to be a family of maps (parameterized by a variable called time) satisfying a certain algebraic property; broadly speaking these two formalisms are equivalent (the operation of turning the latter formalism to what you suggest is called "Currying"; and the opposite is called "unCurrying" (among other things); see also How a group represents the passage of time? and Show that group action is homomorphism to Symmetric group). One point is that often the "two variable" formalism is more convenient to describe analytic properties (like smoothness), whereas the formalism you suggest is more convenient to describe algebraic properties.