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I am interested in vector fields whose Jacobian has orthogonal columns; i.e. if $\mathbf{f}(\cdot):\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a function where $\mathbf{f}(\mathbf{x})=[f_1(\mathbf{x}), f_2(\mathbf{x}),~\dots, f_n(\mathbf{x})]^{\rm T}$, I am looking for all such functions that:

$\forall~\mathbf{x}\in\mathbb{R}^{n}\quad\&\quad 1\leq i,j\leq n:\quad \big(\nabla f_i(\mathbf{x})\big)^{\rm T}\nabla f_j(\mathbf{x}) = \begin{cases}0~&:i\neq j\\g_{i}(\mathbf{x})&:i=j \end{cases}$

A similar question has been asked here. As I understood, in Liouville's theorem for conformal maps all the diagonal elements of the Jacobian $\nabla\mathbf{f}(\mathbf{x})$ are the same. Here, however, I am looking for a generalized case where the diagonal elements are not necessarily the same. Do we have something similar to Liouville's theorem for this case?

Thanks.

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    $\begingroup$ Clearly, there are more examples here than in the case of Liouville's theorem: if $g$ is a conformal map and $f_i(x) = h_i(g_i(x))$ for an arbitrary collection of $h_i : \mathbb{R} \to \mathbb{R}$, then the gradients of $f_i$ are orthogonal. $\endgroup$ – Mateusz Kwaśnicki Jul 8 '20 at 6:48
  • $\begingroup$ Just one remark. Your definition does not imply (is not equivalent to) that the rows be orthogonal. Thus there would be a symmetric question concerning those fields whose Jacobian has orthogonal rows. $\endgroup$ – Denis Serre Jul 13 '20 at 10:37
  • $\begingroup$ @DenisSerre: A good remark, but, in fact, the two problems are equivalent up to computing an inverse function. If one regards $x$ and $f$ as columns of height $n$, the Jacobian $J$ satisfies $df = J\,dx$, and the stated condition is that $J^T\,J$ be diagonal. If $K = J^{-1}$, then $dx = K\,df$, and we see that $$K^T\,K = (J^{-1})^T\,(J^{-1})=( J\,J^T)^{-1,},$$ hence $K^TK$ is diagonal if and only if $J\,J^T$ is diagonal. Thus, $f:\mathbb{R}^n\to\mathbb{R}^n$ satisfies the column condition if and only if the inverse function $f^{-1}:\mathbb{R}^n\to\mathbb{R}^n$ satisfies the row condition. $\endgroup$ – Robert Bryant Sep 30 '20 at 0:25
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You are asking about the subject of orthogonal (coordinate) systems. There is an extensive literature on this subject, in particular by Darboux when $n=3$, and if you search on "triply orthogonal systems", you will see references to Darboux, Eisenhart, etc. plus many more recent references. There are many classical examples.

The equations are underdetermined when $n=2$, determined when $n=3$, and overdetermined when $n>3$, but they are always involutive and 'integrable' (in the sense of integrable systems).

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  • $\begingroup$ Thanks for your response. Actually I'm interested in the case where $n>3$ and the function is invertible. Does the over-determined means that they don't have any solution? $\endgroup$ – oasis93 Jul 18 '20 at 10:19
  • $\begingroup$ @oasis93: There are plenty of invertible solutions for all $n$. For example, if $f_j$ is an invertible function of $x_j$ only for all $j$, you have a solution. Of course, this is not all of the solutions. The involutivity property implies that there exist many solutions locally and describes how to construct them using PDE. Overdetermined means that you can't arbitrarily prescribe initial conditions for the PDE on a hypersurface, say, of the form $\mathbb{R}^{n-1}\subset\mathbb{R}^n$. $\endgroup$ – Robert Bryant Jul 18 '20 at 10:39
  • $\begingroup$ Can you please introduce some references for the general case of n>3? I tried terms such as "over-determined orthogonal coordinate systems" on Google, but it seems there's no exact result. When searching for "orthogonal coordinate systems" almost all the results are for the case of n=3, which is not of interest to me. I would appreciate if you could introduce a book or paper on the general case of n>3 that characterizes the solutions. Huge thanks. $\endgroup$ – oasis93 Sep 29 '20 at 20:50
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    $\begingroup$ @oasis93: Well, if you read French, you can start with Gaston Darboux' 1898 book, Leçons sur les Systèmes Orthogonaux et les Coordonnées Curvilignes, which contains considerable information about the $n$-dimensional case for $n>3$. It covers pretty much everything that was known about the subject before 1900. There is an informative and thorough review of this book (in English) by E. O. Lovett published by the AMS at ams.org/journals/bull/1899-05-04/S0002-9904-1899-00584-6/… $\endgroup$ – Robert Bryant Sep 29 '20 at 23:47

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