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Suppose $ f: \mathbb{R}^3 \to \mathbb{R}$ is a smooth function and $V = \nabla f$ is a smooth vector field. Let $\theta : \mathbb{R} \times \mathbb{R}^3 \to \mathbb{R}^3$ be the integral curves of $V$. It would be very good for my problem if I could construct a chart $\psi = (t,\theta,\varphi)$ with the property that $(f\circ \psi)^{-1}(a) = \{p\} \times [0,2\pi]\times [0,\pi]$ is some vertical two dimensional plane in the domain of $\psi$ for each $a$ in the image of $f$. I would very much like to determine the Laplacian in these coordinates. I have taken an introductory course to smooth manifolds during my master studies, so I have a basic understanding of smooth manifolds. I feel this should be possible for the particular $f$ that I am using, but I am not sure how to do it. (Please let me know if the question is not well-posed, I am a beginner, so that would be helpful.) If you know how it is possible to solve my problem, it would be great if you can provide some details, such that I can read up on it and see if it works.

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You can do it away from the zero of $V$ (i.e. the critical point of $f$).

Let $q \in \mathbb R^3$ be a point where $V(q)\neq 0$. By the Straightening theorem for vector fields, there is an open set $U$ with coordinates $(t, x, y)$ and a local coordinates $\varphi : U \to \mathbb R^3$ so that $V = \frac{\partial }{\partial t}$ in $\varphi(U)$. By shrinking $U$, we assume $U = I \times V$ for some interval $I$. Under this coordinates, the integral curve $\theta$ is given by

$$ \theta (t_0, (t, x, y )) = (t_0+ t, x, y)$$

and after a scaling in the $(x, y)$ direction, this coordinates has the properties you need.

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    $\begingroup$ Great news Arctic Char! Do you know offhand how to get a formula for the Laplacian of these coordinates, or must I go through the proof of the straightening theorem to see if I can get an expression for $\varphi$? $\endgroup$ – Mikkel Rev Nov 29 '19 at 23:30
  • $\begingroup$ I have gone through the proof now. It seems difficult get an explicit formula for the coordinates because I must invert a mean-looking solution of a PDE. Is the easiest way to get the laplacian to use the Voss-Weyl formula? If so, do you know how I can get the metric in such coordinates? Thank you! $\endgroup$ – Mikkel Rev Nov 30 '19 at 0:38
  • $\begingroup$ Well you can represent the Laplacian as the sum of Laplacian (of the level set $f=constant$) and the mean curvature of the level set (together with one more term). Is that what you want? @MikkelRev $\endgroup$ – Arctic Char Nov 30 '19 at 1:18

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