Flows

Lee Smooth Manifolds

Recall the following fundamental theorem for autonomous ODEs: Suppose $U\subset {\mathbb{R}}^n$ is open and $V:U\to {\mathbb{R}}^n$ is smooth. Consider the initial value problem

$$\dot{y}(t)=V(y(t)),y(t_0)=c$$

for arbitrary $t_0\in \mathbb{R}$ and $c\in U$. The following statements hold. (a) Existence: For any $t_0\in \mathbb{R}$ and $x_0\in U$, there exists an open interval $J_0$ containing $t_0$ and an open subset $U_0\subset U$ containing $x_0$ such that for each $c\in U_0$, there is a $C^1$ map $y:J_0\to U$ that solves the IVP. (b) Uniqueness: Any two differentiable solutions to the IVP agree on their common domain. (c) Smoothness: Let $J_0$ and $U_0$ be as in (a), and let $\theta :J_0\times U_0\to U$ be the map defined by $\theta (t,x)=y(t)$ where $y:J_0\to U$ is the unique solution to the IVP with $c=x$. Then $\theta$ is smooth.

Theorem 1: If $M$ is a smooth manifold and $\theta :\mathcal{D}\to M$ is a smooth flow, the infinitesimal generator $V$ of $\theta$ is a smooth vector field, and each curve ${\theta }^{(p)}$ is an integral curve of $V$.

Proof) For the smoothness of $V$, it suffices to show $Vf$ is smooth for all $f\in C^{\infty }(M)$. Note

$$Vf(p)=V_p(f)=d{\theta }_{(0,p)}(\frac{\partial }{\partial t})f=\frac{\partial }{\partial t}(f\circ \theta ){|}_{(0,p)}$$

which is clearly a smooth function of $p$, and thus we are done. Now we show ${\theta }^{(p)}$ is an integral curve. Take any arbitrary $t_0\in {\mathcal{D}}^{(p)}$; it suffices to show $d{\theta }_{(p,t_0)}(\partial /\partial t)=V_{\theta (p,t_0)}$. There must exist some $\epsilon >0$ such that $(t_0-\epsilon ,t_0+\epsilon )\subset {\mathcal{D}}^{(p)}$ and such that $(-\epsilon ,\epsilon )\subset {\mathcal{D}}^{(\theta (p,t_0))}$. Thus, ${\theta }^{(\theta (p,t_0))}(t)={\theta }^{(p)}(t+t_0)$ for $t\in (-\epsilon ,\epsilon )$, implying

$$V_{\theta (p,t_0)}=d{\theta }_0^{(\theta (p,t_0))}(\frac{d}{dt})=d{\theta }_{t_0}^{(p)}(\frac{d}{dt})$$

and completing our proof.

Theorem 2 (Fundamental Theorem on Flows): Let $V$ be a smooth vector field on a smooth manifold $M$. There exists a unique smooth maximal flow $\theta :\mathcal{D}\to M$ whose infinitesimal generator is $V$. This flow has the following properties: (a) For each $p\in M$, ${\theta }^{(p)}:{\mathcal{D}}^{(p)}\to M$ is the unique maximal integral curve of $V$ starting at $p$. (b) If $s\in {\mathcal{D}}^{(p)}$, then ${\mathcal{D}}^{\theta (s,p)}={\mathcal{D}}^{(p)}-s$. (c) For each $t\in \mathbb{R}$, $M_t$ is open in $M$ and ${\theta }_t$ is a diffeomorphism from $M_t$ to $M_{-t}$ with inverse ${\theta }_{-t}$.

Proof) By (a) of the ODE theorem, for any $p\in M$ there exists some integral curve starting at $p$ defined in some neighborhood of $0$. Note that if there are two integral curves $\gamma ,{\gamma }_0$ both defined on some open interval $J\subset \mathbb{R}$ and for some $t_0\in J$ $\gamma (t_0)={\gamma }_0(t_0)$, we have that $\gamma (t)={\gamma }_0(t)$ for all $t\in J$. We quickly prove this claim; let $J^{\prime }=\{t\in J|\gamma (t)={\gamma }_0(t)\}$, which is a closed subset of $J$ containing $t_0$. For any $t^{\prime }\in J^{\prime }$, there exists some $ϵ>0$ such that $\gamma ((t^{\prime }-ϵ,t^{\prime }+ϵ))\subset U$ and ${\gamma }_0((t^{\prime }-ϵ,t^{\prime }+ϵ))\subset U$ for some smooth chart $U\subset M$. Transferring the problem into ${\mathbb{R}}^n$ and applying (b) of the ODE theorem forces $\gamma ={\gamma }_0$ for $t\in (t^{\prime }-ϵ,t^{\prime }+ϵ)$, and thus $(t^{\prime }-ϵ,t^{\prime }+ϵ)\subset J^{\prime }$. Thus $J^{\prime }$ is both open and closed in $J$, forcing $J^{\prime }=J$.

We can now define $\theta$ and $\mathcal{D}$. For any $p\in M$, let ${\mathcal{D}}^{(p)}$ be the union of all open intervals $J$ containing $p$ on which an interval curve starting at $p$ is defined. For any $t\in {\mathcal{D}}^{(p)}$, let $\theta (p,t)=\gamma (t)$ for some integral curve $\gamma$ starting at $p$ and defined on an open interval containing $[0,t]$. This is well-defined by our observation that intersecting integral curves are identical on their common domain. Note we have yet to show that $\mathcal{D}$ is a valid flow domain or that $\theta$ is a valid smooth flow.

Note that property (a) of $\theta$ is trivial by definition. $\theta$ also clearly satisfies the group properties of flows and (b) by the translation lemma. Our remaining tasks are as follows: to prove that $\mathcal{D}$ is open (and thus a valid flow domain), to show $\theta$ is smooth, to show (c), and to prove uniqueness.

We show $\theta$ is smooth and $\mathcal{D}$ is open using one argument. Define $W\subset \mathcal{D}$ to be the collection of points $(t,p)\in \mathcal{D}$ such that $\theta$ is defined and smooth on a product neighborhood of the form $J\times U\subset \mathcal{D}$, where $J$ is an open interval containing $0$ and $U\subset M$ is some neighborhood of $p$. $W$ must be open in $\mathbb{R}\times M$; thus, it suffices to show $W=\mathcal{D}$. By (a) and (c) of the ODE theorem, a neighborhood of $\{0\}\times M$ is in $W$.

FTSOC assume $(\tau ,p_0)\in \mathcal{D}\setminus W$, and for convenience assume $\tau >0$. Consider $t_0=\sup \{{\mathcal{D}}^{(p_0)}\cap W^{(p_0)}\}$; clearly, $\tau \ge t_0$. We will prove $\theta$ is smooth in some product neighborhood of $(t_0,p_0)$, thus creating a contradiction.

Let $q_0=\theta (t_0,p_0)$. By the ODE theorem, there must be some product neighborhood $(-ϵ,ϵ)\times U_0$ of $q_0$ on which $\theta$ is defined and smooth.