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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 $\gamma_{t_1}(p) \circ \gamma_{t_2}(p) = \gamma_{t_1 + t_2}(p)$?

CBBAM
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1 Answers1

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It might be good to keep in mind the earlier discussion Understanding the relationship between local flows and vector fields.

Is this extra information of smooth dependence on initial conditions useful?

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.

[W]hy 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 $\gamma_{t_1}(p) \circ \gamma_{t_2}(p) = \gamma_{t_1 + t_2}(p)$?

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.

Alp Uzman
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  • Thank you for another helpful answer! My confusion initially arose when studying Lie derivatives where it seemed that using an integral curve instead of a local flow would suffice. Based on your comment it seems that idea would be correct in the sense that an integral curve is equivalent to a local flow restricted to a single point? The difference is that local flows allow us to study the time evolution of a collection of points in some neighborhood which would otherwise be cumbersome to do using integral curves, and this is of topological interest. – CBBAM Feb 17 '23 at 05:19
  • I am also interested in the alternate viewpoint you mentioned. If we have a flow defined on some set $U$, instead of viewing it as a collection of single point trajectories/integral curves, would it be correct to picture this as a collection of points that are in some sense "moving together" (hence the flow describes the trajectory of $U$)? – CBBAM Feb 17 '23 at 05:21
  • I'm glad it was useful. Your interpretation of points moving together is accurate. Also this is not only of topological interest. For instance once a volume on the manifold is fixed one can ask if the flow preserves this volume. Infinitesimally this corresponds to the generating vector field being "divergence free". – Alp Uzman Feb 17 '23 at 05:30
  • For Lie derivatives too it might be useful to consider the flow instead of a trajectory; indeed one can consider the Lie derivative $L_X$ along $X$ (or along the flow generated by $X$) as an operator $C^r(M;\mathbb{R})\to C^{r-1}(M;\mathbb{R})$ (or alternatively as an operator acting on the space of sections of some more sophisticated bundle, like tensors). So for $f:M\to \mathbb{R}$ a function one would want to be able to talk about the differentiability of $L_X(f)$ in directions transverse to the flow direction. – Alp Uzman Feb 17 '23 at 05:33
  • If we consider a situation where a neighborhood $U$ is subject to some vector field whose divergence is positive. In the simplest case I am imagining an a fixed volume in space where everything is moving radially outward from some fixed center. In this case, each point must be following a different trajectory, so how would we say that the points in $U$ are all moving together? – CBBAM Feb 17 '23 at 05:51
  • @CBBAM This is where you would use the "two variable" formalism; as long as $U$ is small enough and $t$ is small enough the new set $U_t=F(t,U)$ would be a set diffeomorphic to $U$. In the example you gave, consider a box in the spherical coordinates and how it would spread out under the action of the flow. – Alp Uzman Feb 17 '23 at 06:10
  • I am a little confused on this last point. So when we say that all the elements in $U$ "move together" we don't mean that all points in $U$ follow the same trajectory, but rather that $U = U_0$ is diffeomorphic to $U_t$? – CBBAM Feb 17 '23 at 06:26
  • @CBBAM If $U$ is open in $M$ and $M$ has at least two dimensions, then $U$ would not fit into one trajectory; some points in $U$ would follow different trajectories. Consider an open ball (or rectangle) on the plane under a linear flow, e.g. $t\mapsto [(x,y)\mapsto (e^{t}x,e^{-t}y)]$. – Alp Uzman Feb 17 '23 at 07:40
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    I see, I think I was overthinking it. So we can informally say all the elements in $U$ move together because all points are mapped by the same trajectory, for example, $t\mapsto [(x,y)\mapsto (e^{t}x,e^{-t}y)]$. In such a case the only difference is that each point has a different initial condition. Is that right? – CBBAM Feb 17 '23 at 08:23