7. Flows

7.1Velocity Vector

Let $M$ be a smooth manifold, $c:\mathbb{R}\to M$ a smooth map. ($c$ is called smooth curve.)

定义 7.1.1. $\dot{c}(t)\in T_{c(t)}M$ is defined by$$D_{t}c(1)=D_{t}c\left(\frac{\partial }{\partial t}\right)$$where $T\mathbb{R}=\begin{array}{r}\mathbb{R}\times \mathbb{R}\\ \left(t,\lambda \cdot \frac{\partial }{\partial t}\right)\end{array}$. This is the velocity vector of $c$ at $t$ (at $c(t)$).

例 7.1.2. $M={\mathbb{R}}^n$, then $c(t)=(x_1(t),\dots ,x_n(t))$. $$\dot{c}(t)=({\dot{x}}_1,\dots ,{\dot{x}}_n)=\left(\frac{\partial x_1}{\partial t},\dots ,\frac{\partial x_n}{\partial t}\right)\in T_{c(t)}{\mathbb{R}}^n={\mathbb{R}}^n$$

7.2Global Flows

定义 7.2.1. A (global) flow on a smooth manifold $M$ is a smooth map $$\phi :M\times \mathbb{R}\to M$$satisfying the following properties: $$\begin{array}{rl}\phi (x,0)& =x\\ \phi (\phi (x,t),s)& =\phi (x,t+s)\end{array}\}\forall x\in M,t,s\in \mathbb{R}$$Write $\phi (x,t)={\phi }_t(x)$, then $$\begin{array}{rl}{\phi }_0& ={\mathrm{Id}}_M\\ {\phi }_t\circ {\phi }_s& ={\phi }_{t+s}\end{array}\}\Rightarrow {\phi }_{-t}=({\phi }_t{)}^{-1}$$

Every ${\phi }_t$ is a smooth map $M\to M$ with a smooth inverse, so ${\phi }_t\in \mathrm{Diff}(M)$.

A flow $\phi$ defines a group homomorphism: $\mathbb{R}\to \mathrm{Diff}(M)$.

定义 7.2.2. $\mathfrak{X}(M):=\Gamma (TM)$ is the vector space of vector fields on $M$.

Given a flow $\phi$, we can define $X\in \mathfrak{X}(M)$ by$$X_p=\left(\frac{\partial }{\partial t}{\phi }_t(p)\right){|}_{t=0}=(D_0\phi )\left(\frac{\partial }{\partial t}\right)$$where$$\begin{array}{rl}\phi :\mathbb{R}& \to M\\ t& \mapsto {\phi }_t(p)\end{array}$$If $p={\phi }_s(q)$, then$$\begin{array}{rl}{X}_p& =\left(\frac{\partial }{\partial t}{\phi }_t(p)\right){|}_{t=0}=\left(\frac{\partial }{\partial t}{\phi }_{t+s}(q)\right){|}_{t=0}\\ & =\left(\frac{\partial }{\partial t}{\phi }_s({\phi }_t(q))\right){|}_{t=0}=D_q{\phi }_s(X_q)\end{array}$$

引理 7.2.3. The vector field $X\in \mathfrak{X}(M)$ obtained by differentiating a flow $\phi$ is invariant under $D\phi$.

7.3Local Flows

Let $M$ be a smooth manifold.

定义 7.3.1. A local flow on $M$ is a covering of $M$ by open sets $U_i$ and a family of smooth maps$${\phi }^i:U_i\times (-{\epsilon }_i,{\epsilon }_i)\to M$$s.t. ${\phi }_0^i={\mathrm{Id}}_{U_i}$ and ${\phi }_t^i\circ {\phi }_s^i={\phi }_{t+s}^i$ whenever all $3$ terms are defined.

命题 7.3.2. For every vector field $X\in \mathfrak{X}(M)$, there exists a local flow $\{U_i\mid i\in I\}$, ${\phi }_i$ such that $$\frac{\partial }{\partial t}{\phi }_t^i(p){|}_{t=0}=X_p$$whenever $p\in U_i$.

证明. The statement is local in $M$, so we can work in a chart, so locally in ${\mathbb{R}}^n$. Using coordinates in ${\mathbb{R}}^n$, we need to solve locally a linear system of ODEs with ${\mathcal{C}}^{\infty }$ coefficients. This can be done!

$\square$

If $U_i\cap U_j\ne \varnothing$, then we require ${\phi }_t^i(x)={\phi }_t^j(x)$ for all $x\in U_i\cap U_j$ and $|t|<\min \{{\epsilon }_i,{\epsilon }_j\}$.

Given $X\in \mathfrak{X}(M)$, we can locally integrate $X$ to get a local flow in this sense.

定义 7.3.3. Two local flows are equivalent if their union is also a local flow.

This is an equivalent relation!

命题 7.3.4. There is a one-to-one corresponding between equivalence classes of local flows on $M$ and vector fields $X\in \mathfrak{X}(M)$.

$(U_i,{\phi }^i)$, $i\in I$ $⇝$ $X$ $⇝$ $(V_j,{\phi }^j)$, $j\in J$ equivalent to $(U_i,{\phi }^i)$, $i\in I$.
$X$ $⇝$ $(V_j,{\phi }^j)$, $j\in J$ $⇝$ $X$.

定义 7.3.5. A vector field $X$ is complete if there is a local flow ${\phi }^i:U_i\times \mathbb{R}\to M$ in the corresponding equivalence class.

Under the one-to-one correspondence in the Proposition 7.3.4, complete vector fields give global flows $\phi :M\times \mathbb{R}\to M$, where $\phi (x,t):={\phi }^i(x,t)$ if $x\in U_i$.

命题 7.3.6. If $X\in \mathfrak{X}(M)$ has compact support$$\mathrm{supp}(X):=\overline{\{x\in M\mid X(x)\ne 0\}}$$then it is complete.

证明. Step 1: Consider a local flow $(U_i,{\phi }^i)$ for $X$, $i\in I$. Since the $U_i$ cover $M$, they cover $\mathrm{supp}(X)$. Since $\mathrm{supp}(X)$ is compact, there exist finitely many $U_i$, say $U_1,\dots ,U_k$, such that $\mathrm{supp}(X)\subset \bigcup _{i=1}^k{U}_i$.

Let $U_0:=M\setminus \mathrm{supp}(X)\underset{\text{open}}{\subset }M$.

Define $\begin{array}{rl}{\phi }^0:U_0\times \mathbb{R}& \to M\\ (x,t)& \mapsto x\end{array}$, $\forall x\in U_0,t\in \mathbb{R}$.
$U_0,U_1,\dots ,U_k$ form a covering of $M$, and the pair $(U_i,{\phi }^i)$, $i\in \{0,\dots ,k\}$ are a local flow for $X$. Set $\epsilon :=\min \{{\epsilon }_1,\dots ,{\epsilon }_k\}>0$.

${\phi }_t(x)={\phi }_t^i(x)$ is defined for all $x\in M$ and all $|t|<\epsilon$.

Step 2: Let $X$ be any vector field which admits a local flow $(U_i,{\phi }^i)$ defined for all times $|t|<\epsilon$. Then we can define ${\phi }_{Nt}^i(x):=\underset{N\text{times for all}N\in \mathbb{N},|t|<\epsilon }{\underbrace{{\phi }_t^i\circ \cdots \circ {\phi }_t^i(x)}}$.

${\phi }_{s+t}={\phi }_s\circ {\phi }_t$ whenever both are defined.

$\square$

推论 7.3.7. If $M$ is compact, then all $X\in \mathfrak{X}(M)$ are complete.

例 7.3.8. Compact support is sufficient for completeness, but not necessary.
$\frac{\partial }{\partial x_1}\ne 0$ everywhere

例 7.3.9. $M={\mathbb{R}}^n\setminus \{0\}$, $X=\frac{\partial }{\partial x_1}\ne 0$ everywhere

If $p=(-s,0,\dots ,0)$, $s>0$, then $\phi (p,t)$ is not defined for $t⩾s$.