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The flow-box theorem

Statement

Let $X$ be a smooth vector field on $M$ , and let $p \in M$ satisfy $X_{p} \neq 0$ . Then there are local coordinates $(u^{1}, \dots, u^{n})$ near $p$ in which

$X = \frac{\partial}{\partial u^{1}} .$

Facts & Assumptions

Given: A smooth vector field $X$ and a point $p$ with $X_{p} \neq 0$ .

[L1]

The maximal flow of $X$ is smooth on an open domain (The fundamental theorem on flows).

[L2]

The manifold inverse function theorem turns a map with invertible differential into a local diffeomorphism (The smooth inverse function theorem on manifolds).

[L3]

Time- $t$ flow maps are diffeomorphisms between open domains (Time-t flow maps are diffeomorphisms between open domains).

Proof

technique · direct

1.1

Choose a chart around $p$ in which the first coordinate component of $X_{p}$ is nonzero, and let $S$ be the codimension-one slice where that first coordinate is constant. Then $T_{p} M = R X_{p} \oplus T_{p} S$ .

given

2.1

Let $Φ$ be the maximal flow of $X$ and define $F (t, q) : = Φ_{t} (q)$ for $(t, q)$ near $(0, p)$ with $q \in S$ . By [L1], $F$ is smooth. Its differential at $(0, p)$ sends the time direction to $X_{p}$ and sends $T_{p} S$ identically into itself, so $d F_{(0, p)}$ is an isomorphism by step 1.1.

L1step 1.1

3.1

Applying [L2] to $F$ at $(0, p)$ gives local coordinates $(u^{1}, \dots, u^{n})$ in which $F$ becomes the identity on an open set of $R \times R^{n - 1}$ . In those coordinates, the flow translates the first coordinate, and therefore its generating vector field is $\partial / \partial u^{1}$ .

L2step 2.1

4.1

Hence every nonvanishing point of a smooth vector field has a neighbourhood in which the field is straightened to a coordinate vector field.

step 3.1L3∎

Depends on

Used by

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)