Understanding the relationship between local flows and vector fields

Question

I am studying from Tu's Introduction to Manifolds, which makes the following two claims:

1. To each flow line is an integral curve of a vector field.
2. Every smooth vector field has a local flow about any point.

I am having trouble understanding the two statements above. In case I am thinking about something incorrectly, I have summarized my current understanding below.

Suppose we have a smooth manifold $M$ and a smooth vector field $X$ on $M$. We can interpret $X$ as assigning a tangent vector to each point in $M$. The way I visualize this is similar to how we draw slope fields in elementary ODEs.

If $p \in U$ where $U$ is open in $M$, then there exists a function $F: (-\epsilon, \epsilon) \times W \rightarrow U$, where $W \subset U$ is an open neighborhood of $p$, such that $$\frac{\partial F}{\partial t}(t,q) = g(F(t,q))$$ and $g$ corresponds to the coefficient of the vector field $X$. Using the notation $F_t(q) = F(t,q)$, we say that $F$ is a local flow generated by a vector field $X$ and $F_t(q)$ is a flow line.

Also, from my understanding, an integral curve is a curve $c: [a,b] \rightarrow M$ such that $c'(t) = X_{c(t)}$. The way I think about integral curves is that the velocity of the curve at a point is equal to the tangent vector given by $X$ at that point.

If my understanding is correct, then a flow line is a curve which starts at a point $q$ and evolves in time by following the tangent vectors given by $X$. Is this why we say that every flow line is an integral curve of $X$?

For the second point, a local flow is not described by a particular point. If I am not mistaken, it gives us the "new position" in $M$ for some point $q$ and time $t$ governed by $X$. So how do we talk about a local flow at any point? Does he mean every smooth vector field has a local flow line about any point?

Answer 1

If my understanding is correct, then a flow line is a curve which starts at a point $q$ and evolves in time by following the tangent vectors given by $X$. Is this why we say that every flow line is an integral curve of $X$?

Yes, this is one of the generalized senses of what is called an integral in single variable calculus. Further, your understanding is correct, perhaps modulo the minor point that the $\epsilon$ bound in the first argument of $F$ may depend on the second argument $p$. Thus a more accurate statement would be to say "for any point $p$, there is a positive number $\epsilon_p$ and a neighborhood $W_p$ of $p$ ...". Assembling $]-\epsilon_p,\epsilon_p[\times W_p$'s as $p$ varies we get an open set $\mathfrak{D}(X)\subseteq \mathbb{R}\times M$ that contains the subset $\{0\}\times M$ of initial conditions and we may write $F:\mathfrak{D}(X)\to M$ (see e.g. the discussion at The proof of the fact that $[v,w]=0\;\Rightarrow\;\exp(\varepsilon v) \exp(\theta w)x=\exp(\theta w)\exp(\varepsilon v)x $ for more details).

For the second point, a local flow is not described by a particular point. If I am not mistaken, it gives us the "new position" in $M$ for some point $q$ and time $t$ governed by $X$. So how do we talk about a local flow at any point? Does he mean every smooth vector field has a local flow line about any point?

The expressions involved ("local flow", "local flow line") are used in closely related but varying senses in the literature (in my opinion your interpretation can be taken as correct). Roughly speaking, the word "flow" emphasizes the fact that not only do we have curves $t\mapsto F(t,p)$ but that there is an algebraic relation satisfied (the so-called group property): $F(t+s,p)=F(t,F(s,p))$, but only insofar as the two sides of this equation makes sense (explicitly, if $s,t,s+t\in ]-\epsilon_p,\epsilon_p[$).

As for how to make sense of a local flow at a point $p$, one can e.g. think of it as the restriction of $F$ above to some rectangular open subset $]-\eta,\eta[\times U$ of $\mathfrak{D}(X)$ containing $(0,p)$ (sometimes such open subsets are called "flow boxes", at least when $X(p)\neq0$). Alternatively (assuming $X(p)\neq0$), one can consider the local flow line $\mathcal{O}_{p,\text{loc}}$ and take the "local flow at $p$" to mean the specific parameterization $F(\bullet,p):]-\epsilon_p,\epsilon_p[\to \mathcal{O}_{p,\text{loc}}$ with the emphasis that this parameterization satisfies the algebraic relation mentioned above (note that $\mathcal{O}_{p,\text{loc}}$ is a $1$-dimensional embedded submanifold of $M$, if $\epsilon_p$ is taken small enough)).