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.

areequivalent 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. $\endgroup$