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\begin{document}

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\section*{flow of a vector field}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{traditional_definition}{Traditional definition}\dotfill \pageref*{traditional_definition} \linebreak
\noindent\hyperlink{SyntheticDefinition}{Synthetic definition}\dotfill \pageref*{SyntheticDefinition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Given a [[tangent vector field]] on a [[differentiable manifold]] $X$ then its \emph{flow} is the [[group]] of [[diffeomorphisms]] of $X$ that lets the points of the manifold ``flow along the vector field'' hence which sends them along \emph{flow lines} (integral curvs) that are tangent to the vector field.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{traditional_definition}{}\subsubsection*{{Traditional definition}}\label{traditional_definition}

Throughout, let $X$ be a [[differentiable manifold]] and let $v \in \Gamma(T X)$ be a continuously differentiable [[vector field]] on $X$ (i.e. of class $C^1$).

\begin{defn}
\label{IntegralCurve}\hypertarget{IntegralCurve}{}
\textbf{(integral curves/flow lines)}

An \emph{integral curve} or \emph{flow line} of the vector field $v$ is a [[differentiable function]] of the form

\begin{displaymath}
\gamma
    \;\colon\;
    U
    \longrightarrow
  X
\end{displaymath}
for $U \subset \mathbb{R}$ an [[open interval]] with the property that its [[tangent vector]] at any $t \in U$ equals the value of the vector field $v$ at the point $\gamma(t)$:

\begin{displaymath}
\underset{t \in U}{\forall}
  \left(
     d \gamma_t = v_{\gamma(t)}
  \right)
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{FlowOfAVectorField}\hypertarget{FlowOfAVectorField}{}
\textbf{(flow of a vector field)}

A \emph{global flow} of $v$ is a function of the form

\begin{displaymath}
\Phi \;\colon\; X \times \mathbb{R} \longrightarrow X
\end{displaymath}
such that for each $x \in X$ the function $\phi(x,-) \colon \mathbb{R} \to X$ is an integral curve of $v$ (def. \ref{IntegralCurve}).

A \emph{flow domain} is an open subset $O \subset X \times \mathbb{R}$ such that for all $x \in X$ the intersection $O \cap \{x\} \times \mathbb{R}$ is an [[open interval]] containing $0$.

A \emph{flow} of $v$ on a flow domain $O \subset X \times \mathbb{R}$ is a differentiable function

\begin{displaymath}
X \times \mathbb{R} \supset O \overset{\phi}{\longrightarrow} X
\end{displaymath}
such that for all $x \in X$ the function $\phi(x,-)$ is an integral curve of $v$ (def. \ref{IntegralCurve}).

\end{defn}
\begin{defn}
\label{CompleteVectorField}\hypertarget{CompleteVectorField}{}
\textbf{(complete vector field)}

The vector field $v$ is called a \emph{complete vector field} if it admits a global flow (def. \ref{FlowOfAVectorField}).

\end{defn}
\hypertarget{SyntheticDefinition}{}\subsubsection*{{Synthetic definition}}\label{SyntheticDefinition}

In [[synthetic differential geometry]] a [[tangent vector field]] is a morphism $v \colon X \to X^D$ such that

\begin{displaymath}
\itexarray{
    && X^D 
    \\
    & {}^{\mathllap{v}}\nearrow & \downarrow^{\mathrlap{X^{\ast \to D}}}
    \\
    X &=& X
  }
\end{displaymath}
The [[internal hom]]-[[adjunct]] of such a morphism is of the form

\begin{displaymath}
\tilde v \;\colon\; D \longrightarrow X^X
  \,.
\end{displaymath}
If $X$ is sufficiently nice (a [[microlinear space]] should be sufficient) then this morphism factors through the internal [[automorphism group]] $\mathbf{Aut}(X)$ inside the internal [[endomorphisms]] $X^X$

\begin{displaymath}
\tilde v \;\colon\; D \longrightarrow \mathbf{Aut}(X) \hookrightarrow X^X
  \,.
\end{displaymath}
Then a group homomorphism

\begin{displaymath}
\phi_v \;\colon\; (R,+) \longrightarrow \mathbf{Aut}(X)
\end{displaymath}
with the property that restricted along any of the affine inclusions $D \hookrightarrow \mathbb{R}$ it equals $\tilde v$

\begin{displaymath}
\itexarray{
     D &\hookrightarrow& \mathbb{R}
     \\
     & {}_{\mathllap{\tilde v}}\searrow & \downarrow^{\mathrlap{\phi}}
     \\
     && \mathbf{Aut}(X) &\hookrightarrow& X^X
  }
\end{displaymath}
is a \emph{flow} for $v$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{prop}
\label{}\hypertarget{}{}
Let $\phi$ be a global flow of a vector field $v$ (def. \ref{FlowOfAVectorField}). This yields an [[action]] of the additive group $(\mathbb{R},+)$ of [[real numbers]] on the [[differentiable manifold]] $X$ by [[diffeomorphisms]], in that

\begin{itemize}%
\item $\phi_v(-,0) = id_X$;


\item $\phi_n(-,t_2) \circ \phi_v(-,t_1) = \phi_v(-, t_1 + t_2)$;


\item $\phi_v(-,-t) = \phi_v(-,t)^{-1}$.



\end{itemize}
\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
\textbf{(fundamental theorem of flows)}

Let $X$ be a [[smooth manifold]] and $v \in \Gamma(T X)$ a smooth [[vector field]]. Then $v$ has a unique maximal flow (def. \ref{FlowOfAVectorField}).

This unique flow is often denoted $\phi_v$ or $\exp(v)$ (see also at \emph{[[exponential map]]}).

\end{prop}
e.g. \hyperlink{Lee}{Lee, theorem 12.9}

\begin{prop}
\label{}\hypertarget{}{}
Let $X$ be a [[compact topological space|compact]] [[smooth manifold]]. Then every smooth [[vector field]] $v \in \Gamma(T X)$ is a complete vector field (def. \ref{CompleteVectorField}) hence has a global flow (def. \ref{FlowOfAVectorField}).

\end{prop}
e.g. \hyperlink{Lee}{Lee, theorem 12.12}

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item John Lee, chapter 12 ``Integral curves and flows'' of \emph{Introduction to smooth manifolds} (\href{http://webmath2.unito.it/paginepersonali/sergio.console/lee.pdf}{pdf})

\end{itemize}
[[!redirects flows of a vector field]]

[[!redirects flow]] [[!redirects flows]] [[!redirects flow line]] [[!redirects flow lines]]

[[!redirects integral curve]] [[!redirects integral curves]]



\end{document}