Local coordinates near point such that $X= \partial_1$ is relative to those coordinates, vector field with isolated s.t. coordinates do not exist?

Question

Let $M$ be a finite-dimensional smooth manifold, and let $X$ be a smooth vector field on $M$. Let $X(p) \neq 0$ for some $p \in M$.

• How do I show that I can find local coordinates near $p$ such that $X = \partial_1$ relative to these coordinates?
• What is an example of a vector field with an isolated zero for which such coordinates do not exist?

Answer 1

How do I show that I can find local coordinates near $p$ such that $X = \partial_1$ relative to these coordinates?

We may assume that $X$ is a vector field in $U \subset \mathbb{R}^n$ and $X(0) \neq 0$ where $p$ is $0$ in these local coordinates. Make a linear change of variables so that $X(0) = \partial_1$. The map$$(x_1, x_2, \dots, x_n) \mapsto \Phi_{x_1} (x_2, x_2, \dots, x_n)$$is a diffeomorphism near the origin; indeed, all we have to do is cite the Inverse Function Theorem, since the aforementioned map written down has the identity as its derivative at the point $0$.

What is an example of a vector field with an isolated zero for which such coordinates do not exist?

An example of a vector field for which it is not possible to straighten is $x \partial_y - y \partial_x$ in $\mathbb{R}^2$. Indeed, in any neighborhood of $0$, we can not straighten the vector field even if we delete the point $0$ because the flow trajectories are circles.