flow of a vector field


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



Given a tangent vector field on a differentiable manifold XX then its flow is the group of diffeomorphisms of XX 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.


Traditional definition

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


(integral curves/flow lines)

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

γ:UX \gamma \;\colon\; U \longrightarrow X

for UU \subset \mathbb{R} an open interval with the property that its tangent vector at any tUt \in U equals the value of the vector field vv at the point γ(t)\gamma(t):

tU(dγ t=v γ(t)). \underset{t \in U}{\forall} \left( d \gamma_t = v_{\gamma(t)} \right) \,.

(flow of a vector field)

A global flow of vv is a function of the form

Φ:X×X \Phi \;\colon\; X \times \mathbb{R} \longrightarrow X

such that for each xXx \in X the function ϕ(x,):X\phi(x,-) \colon \mathbb{R} \to X is an integral curve of vv (def. 1).

A flow domain is an open subset OX×O \subset X \times \mathbb{R} such that for all xXx \in X the intersection O{x}×O \cap \{x\} \times \mathbb{R} is an open interval containing 00.

A flow of vv on a flow domain OX×O \subset X \times \mathbb{R} is a differentiable function

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

such that for all xXx \in X the function ϕ(x,)\phi(x,-) is an integral curve of vv (def. 1).


(complete vector field)

The vector field vv is called a complete vector field if it admits a global flow (def. 2).

Synthetic definition

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

X D v X *D X = X \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

v˜:DX X. \tilde v \;\colon\; D \longrightarrow X^X \,.

If XX is sufficiently nice (a microlinear space should be sufficient) then this morphism factors through the internal automorphism group Aut(X)\mathbf{Aut}(X) inside the internal endomorphisms X XX^X

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

Then a group homomorphism

ϕ v:(R,+)Aut(X) \phi_v \;\colon\; (R,+) \longrightarrow \mathbf{Aut}(X)

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

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

is a flow for vv.



Let ϕ\phi be a global flow of a vector field vv (def. 2). This yields an action of the additive group (,+)(\mathbb{R},+) of real numbers on the differentiable manifold XX by diffeomorphisms, in that

  • ϕ v(,0)=id X\phi_v(-,0) = id_X;

  • ϕ n(,t 2)ϕ v(,t 1)=ϕ v(,t 1+t 2)\phi_n(-,t_2) \circ \phi_v(-,t_1) = \phi_v(-, t_1 + t_2);

  • ϕ v(,t)=ϕ v(,t) 1\phi_v(-,-t) = \phi_v(-,t)^{-1}.


(fundamental theorem of flows)

Let XX be a smooth manifold and vΓ(TX)v \in \Gamma(T X) a smooth vector field. Then vv has a unique maximal flow (def. 2).

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

e.g. Lee, theorem 12.9


Let XX be a compact smooth manifold. Then every smooth vector field vΓ(TX)v \in \Gamma(T X) is a complete vector field (def. 3) hence has a global flow (def. 2).

e.g. Lee, theorem 12.12


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

Last revised on June 12, 2018 at 15:41:51. See the history of this page for a list of all contributions to it.