The flow-box theorem

The flow-box theorem, or domain-straightening theorem, states that for a vector field $X$, there are local coordinates such that $X = \partial_1$, if $X \neq 0$.

Theorem : For a uniformly Lipschitz-continuous vector field $X \in \Gamma(TM)$ that is non-singular at $p$ (ie, $X(p) \neq 0$, there exists a neighbourhood $U \ni p$ for which there exists a coordinate chart in which, for any $C^1$ function,

\begin{equation} X[f] = \partial_1 f(x_1, x_2, \ldots) \end{equation}

ie, $X = \partial_1$.

Proof : Take some coordinate system $\phi(p) = (y_1, y_2, \ldots)$ in an open set of $p$ small enough that $X \neq 0$. In this coordinate chart,

\begin{equation} X = X^\mu(y) \partial^y_\mu \end{equation}

Using the Gram-Schmidt process, we can find another basis at $T_pM$ such that $X^{\mu'} = (1, 0, \ldots)$. Consider now the integral curves of $X$ at $p$, $\dot{\gamma}(\lambda) = X(\gamma(\lambda))$.

As our vector field is uniformly Lipschitz-continuous, for a given there is some neighbourgood $[-\varepsilon, \varepsilon]$ for which there is a unique such curve.