I am interested in functions 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.

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.