nLab flow of a vector field

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 curves) that are tangent to the vector field.

Integral curves generalize to integral sections for multivector fields. See there for more.

Definition

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 $C^1$).

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).

e.g. Lee, theorem 12.9

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. ).

e.g. Lee, theorem 12.12

References

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