Consider the following definition of conjugate nets in arbitrary dimensions (in the following $3 \leq N$ and $2\leq m < N$).
Definition: An immersion $f: \mathbb{R}^m \longrightarrow \mathbb{R}^N$ is called an $m$-dimensional conjugate net in $\mathbb{R}^N$ if at every point $p$ of $\mathbb{R}^m$ and for all pairs $1 \leq i \neq j \leq m$ one has $\partial_i\partial_j f \in \textrm{span}(\partial_i f, \partial_j f)$ where by $\partial_i f$ we mean $\frac{\partial f}{\partial x_i}$.
In the book Discrete Differential Geometry, the author claims that a generic surface in $\mathbb{R}^4$ has a unique conjugate net and in higher dimensions such a parametrization does not need to exist at all. Up to my knowledge the author does not provide any proof or reference to the claim (so I suspect this must be a classical result). Therefore, my questions are
- How to prove the claim (a proper reference would suffice too),
- Is it possible to have a geometric description for the case of $\mathbb{R}^4$? Similar to what already exists for the Dupin Indicatrix of regular surfaces in $\mathbb{R}^3$ (see conjugate diameter of conics)?