Time-Dependent Vector Fields and Flows

Subjects: Differential Geometry, Ordinary Differential Equations
Links: Integral Curves, Flows and Flowouts on Smooth Manifolds, Existence and Uniqueness of Solutions to Systems of Differential Equations

Let $M$ be a smooth manifold. A time-dependent vector field on $M$ is a continuous map $V:J\times M\to TM$, where $J\subseteq \mathbb{R}$ is an interval, such that $V(t,p)\in T_{p}M$ for each $(t,p)\in J\times M$. This means that for each $t\in J$, the map $V_t:M\to TM$ is defined by $V_t(p)=V(t,p)$ is a vector field on $M$. If $V$ is a time-dependent vector field on $M$, an integral curve of $V$ is a differentiable curve $\gamma :J_0\to M$, where $J_0$ is an interval contained in $J$, such that $$\gamma'(t) = V(t, \gamma(t)), \quad \text{for all }t\in J_0.$$
Every ordinary vector field $X\in \mathfrak{X}(M)$ determines a time-dependent vector field defined on $\mathbb{R}\times M$, just by setting $V(t,p)=X_p$.

A time-dependent vector field might not generate flow, because two integral curves starting at the same point but different times might flow different paths, whereas all integral curves of a flow through a given point have the same image.

Fundamental Theorem on Time-Dependent Flows: Let $M$ be a smooth manifold, let $J\subseteq \mathbb{R}$ be an open interval, and let $V:J\times M\to TM$ be a smooth time-dependent vector field on $M$. There exists an open subset $\mathcal{E}\subseteq J\times J\times M$ and a smooth map $\psi :\mathcal{E}\to M$ called the time-dependent flow of $V$, with the following properties:

• For each $t_0\in J$ and $p\in M$ the set ${\mathcal{E}}^{(t_0,p)}=\{t\in J\mid (t,t_0,p)\in \mathcal{E}\}$ is an open interval containing $t_0$, and the smooth curve ${\psi }^{(t_0,p)}:{\mathcal{E}}^{(t_0,p)}\to M$ defined by ${\psi }^{(t_0,p)}(t):=\psi (t,t_0,p)$ is the unique maximal integral curve of $V$ with initial condition ${\psi }^{(t_0,p)}(t_0)=p$.
• If $t_1\in {\mathcal{E}}^{(t_0,p)}$ and $q={\psi }^{(t_0,p)}(t_1)$, then ${\mathcal{E}}^{(t_1,q)}={\mathcal{E}}^{(t_0,p)}$ and ${\psi }^{(t_1,q)}={\psi }^{(t_0,p)}$.
• For each $(t_1,t_0)\in J\times J$, the set $M_{t_1,t_0}:=\{p\in M\mid (t_1,t_0,p)\in \mathcal{E}\}$ is an open set in $M$, and the map ${\psi }_{t_1,t_0}:M_{t_1,t_0}\to M$ defined by ${\psi }_{t_1,t_0}(p):=\psi (t_1,t_0,p)$ is a diffeomorphism from $M_{t_1,t_0}$ onto $M_{t_0,t_1}$ with inverse ${\psi }_{t_0,t_1}$.
• If $p\in M_{t_1,t_0}$ and ${\psi }_{t_1,t_0}(p)\in M_{t_2,t_1}$, then $p\in M_{t_2,t_0}$ and $$(\psi_{t_2, t_1}\circ \psi_{t_1, t_0}) (p) = \psi_{t_2, t_0} (p). $$
  Prop: $M$ is a compact smooth manifold and $V:J\times M\to TM$ is a smooth time-dependent vector field on $M$. Then the domain of the time-dependent flow of $V$ is all of $J\times J\times M$.

This came to me as the analogous idea of a complete vector field, like the flow is as global as it can be.

Def: Let $M$ be a smooth manifold. A smooth isotopy of $M$ is a smooth map $H:M\times J\to M$, where $J\subseteq \mathbb{R}$ is an interval, such that for each $t\in J$, the map $H_t:M\to M$ defined by $H_t(p)=H(p,t)$ is diffeomorphism. In particular if $J$ is the unit interval, then $H$ is a homotopy from $H_0$ to $H_1$ through diffeomorphism.

Prop: Suppose $J\subseteq \mathbb{R}$ is an open interval and $H:M\times J\to M$ is a smooth isotopy. Then the map $V:J\times M\to TM$ defined by$$V(t, p) :=\frac{\partial}{\partial t} H(H_t^{-1}(p), t) $$is a smooth time-dependent vector field on $M$, whose time dependent flow is given by $\psi (t,t_0,p)=(H_t\circ H_{t_0}^{-1})(p)$ with domain $J\times J\times M$.

Prop: Suppose $J$ is an open interval and $V:J\times M\to M$ is a smooth time-dependent vector field on $M$ whose time-dependent flow is defined on $J\times J\times M$. For any $t_0\in J$, the map $H:M\times J\to M$ defined $H(t,p):=\psi (t,t_0,p)$ is smooth isotopy of $M$,

This again feels like the analogous theorem for global flows.