# nLab flow of a vector field

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

Given a tangent vector field on a differentiable manifold $X$ then its 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 flow lines (integral curvs) that are tangent to the vector field.

## 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 $C^1$).

###### Definition

(integral curves/flow lines)

An integral curve or flow line of the vector field $v$ is a differentiable function of the form

$\gamma \;\colon\; U \longrightarrow X$

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 $\gamma(t)$:

$\underset{t \in U}{\forall} \left( d \gamma_t = v_{\gamma(t)} \right) \,.$
###### Definition

(flow of a vector field)

A global flow of $v$ is a function of the form

$\Phi \;\colon\; X \times \mathbb{R} \longrightarrow X$

such that for each $x \in X$ the function $\phi(x,-) \colon \mathbb{R} \to X$ is an integral curve of $v$ (def. ).

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

$X \times \mathbb{R} \supset O \overset{\phi}{\longrightarrow} X$

such that for all $x \in X$ the function $\phi(x,-)$ is an integral curve of $v$ (def. ).

###### Definition

(complete vector field)

The vector field $v$ is called a complete vector field if it admits a global flow (def. ).

### Synthetic definition

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

$\array{ && X^D \\ & {}^{\mathllap{v}}\nearrow & \downarrow^{\mathrlap{X^{\ast \to D}}} \\ X &=& X }$

The internal hom-adjunct of such a morphism is of the form

$\tilde v \;\colon\; D \longrightarrow X^X \,.$

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 $X^X$

$\tilde v \;\colon\; D \longrightarrow \mathbf{Aut}(X) \hookrightarrow X^X \,.$

Then a group homomorphism

$\phi_v \;\colon\; (R,+) \longrightarrow \mathbf{Aut}(X)$

with the property that restricted along any of the affine inclusions $D \hookrightarrow \mathbb{R}$ it equals $\tilde v$

$\array{ D &\hookrightarrow& \mathbb{R} \\ & {}_{\mathllap{\tilde v}}\searrow & \downarrow^{\mathrlap{\phi}} \\ && \mathbf{Aut}(X) &\hookrightarrow& X^X }$

is a flow for $v$.

## Properties

###### Proposition

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

• $\phi_v(-,0) = id_X$;

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

• $\phi_v(-,-t) = \phi_v(-,t)^{-1}$.

###### Proposition

(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. ).

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

###### Proposition

Let $X$ be a compact smooth manifold. Then every smooth vector field $v \in \Gamma(T X)$ is a complete vector field (def. ) hence has a global flow (def. ).

• John Lee, chapter 12 “Integral curves and flows” of Introduction to smooth manifolds (pdf)